Effect of Angular Velocity of Inner Cylinder on Laminar Flow through ...

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 02<strong>Effect</strong> <strong>of</strong> <strong>Angular</strong> <strong>Velocity</strong> <strong>of</strong> <strong>Inner</strong> <strong>Cylinder</strong> onLaminar Flow through Eccentric Annular CrossSection PipeRessan Faris Hamd *Department <strong>of</strong> Mechanical Engineering, Kufa University, IraqCorresponding author; E-mail address: Ressan@yahoo.comAbstract-- This is study investigated the effect increasing theangular velocity on the axial velocity pr<strong>of</strong>ile and pressure gradientfor steady state, three dimensional, incompressible, isothermal, andlaminar flow <strong>of</strong> a Newtonian fluid through eccentric annular duct.The results are presented for angular velocity range (0, 50,100, 150, 200, and 250) rpm at axial Reynolds number 200 based onbulk axial velocity with radius ratio 0.5 where FLUENT s<strong>of</strong>twareversion (6.2) is used to solve continuity and momentum equations.The results <strong>of</strong> axial and tangential velocity for four sectors werefound to be in good agreement with other previously publishedresearch at eccentricity are 0.2 and 0.5.The results show that the axial velocity varies with varyingin rotational speed <strong>of</strong> inner cylinder and the axial pressure gradientincreases with increasing in rotational speed. Also when innercylinder is rotated, the axial velocity pr<strong>of</strong>ile become nonaxisymmetryaround center <strong>of</strong> radial gap <strong>of</strong> annular pipe ateccentricity <strong>of</strong> 0.2 and 0.4.Index Term-- annular, rotate cylinder, laminar, eccentricNOMENCLATURELatinDescriptionsymbolsdisplacement <strong>of</strong> inner cylinder axis fromeouter cylinder axis (m)radial distance from axis <strong>of</strong> inner cylinderr(m)p' Axial pressure gradient (Pa/m)R I outer radius <strong>of</strong> inner cylinder (m)R O inner radius <strong>of</strong> outer cylinder (m)Re axial Reynolds number 2ρUδ/μ (--)non-dimensional axial component <strong>of</strong>uvelocity (--)non-dimensional tangential component <strong>of</strong>vvelocity (--)u axial component <strong>of</strong> velocity (m/s)v tangential component <strong>of</strong> velocity (m/s)U bulk axial velocity (m/s)non-dimensional radial component <strong>of</strong>wvelocity (m/s)Cartesian coordinate in horizontal directionx(m)Non-dimensional value <strong>of</strong> distance fromyouter wall (m)GreeksymbolsξDescriptionNon-dimensional distance from wall <strong>of</strong>inner cylinder (--)κ Radius ratio R I /R O (--)ε Eccentricity e/δ (--)ρ density <strong>of</strong> the air (kg/m 3 )Azimuthal location with respect to innerϕcylinder (degree)δ Annular gap width R O - R I (m)ω angular velocity <strong>of</strong> inner cylinder (rpm)µ dynamic viscosity <strong>of</strong> the fluid (kg/m.s)v kinematic viscosity <strong>of</strong> the fluid (m 2 /s)symbolsAbbreviationsCFD Computational Fluid DynamicI. INTRODUCTIONThe classical Couette flow problem consists <strong>of</strong>infinitely long concentric cylinders and an incompressibleNewtonian fluid between them. For a system with a rotatinginner cylinder and a stationary outer cylinder, the fluid flowwill pass on stable circular Couette flow and steadyaxisymmetric Taylor vortex flow for the value <strong>of</strong> Taylornumber Ta (or Reynolds number Re) less than and greater thana critical value Ta c (or Re c ) respectively [1].The flow device <strong>of</strong> Taylor–Couette flow comprisingconcentric rotating cylinders is widely used in many industrialand research processes found in chemical, mechanical andnuclear engineering. The device can be only one cylinderrotating and the other at rest, or two cylinders rotating in thesame or counter directions. The accurate calculation <strong>of</strong> theflow property is important even from the standpoint <strong>of</strong> thenormal operation <strong>of</strong> the device. The distribution <strong>of</strong> energy lossin the device may greatly influence the industrial process <strong>of</strong>mixing, diffusion, heat transfer, and flow stabilities, etc. [2].C. H. Ataide et al [3]: (2003) analyzed the fluid flow inannular regions is an area <strong>of</strong> great interest in the petroleumindustry, both in the drilling and in the artificial rising <strong>of</strong> thepetroleum. Through the numerical simulation, using thecomputer fluid dynamics commercial code (CFD). Theyinvestigated flow <strong>of</strong> non-Newtonian fluids flow through theannular space formed by two tubes in concentric and eccentricMay 2013ATE-80302020©Asian-Transactions39

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 02arrangements <strong>of</strong> a horizontal system. They studiedcontemplates the prediction <strong>of</strong> effects <strong>of</strong> viscosity,eccentricity, flow and inner axis rotation velocity over thetangential and axial velocity pr<strong>of</strong>iles and over thehydrodynamic losses, considering that these two variables arerelevant in the understanding <strong>of</strong> the drilling fluids flow. Theyevaluated the performance by using numerical technique, andcompared the obtained results with other reported works.C. Shu et al [1]: (2004) used the differential quadrature (DQ)method to simulate the eccentric Couette–Taylor vortex flowin an annulus between two eccentric cylinders with rotatinginner cylinder and stationary outer cylinder. Their approachcombining the SIMPLE (semi-implicit method for pressurelinked equations). They proposed to solve the time-dependent,three-dimensional incompressible Navier–Stokes equations inthe primitive variable form. They obtained eccentric steadyCouette–Taylor flow patterns from the solution <strong>of</strong> threedimensionalNavier–Stokes equations. For steady eccentricTaylor vortex flow, detailed flow patterns were obtained andanalyzed. The effect <strong>of</strong> eccentricity on the eccentric Taylorvortex flow pattern was also studied.Escudier et al [4]: (2000) showed that there is excellentagreement between their calculations for a Newtonian fluidand experimental data for annuli with 20%, 50%, and 80%eccentricity. Their results are reported for calculations <strong>of</strong> theflow field, wall shear stress distribution and friction factor fora range <strong>of</strong> values <strong>of</strong> eccentricity, radius ratio, and Taylornumber. They showed the axial component <strong>of</strong> velocity isdirectly affected by the radial/tangential velocity field androtation <strong>of</strong> the inner cylinder is found to have a stronginfluence on the axial velocity distribution. As the Taylornumber is increased the friction factor for high values <strong>of</strong> ε >0.9 increases.Escudier et al. [5]: (2002) evaluated the effect <strong>of</strong> internalcylinder rotation on the laminar flow <strong>of</strong> non-Newtonian fluidsin an eccentric annular region, compared their numericalsimulations <strong>of</strong> two-dimensional flow against experimentaldata. Their comparison <strong>of</strong> simulated velocity pr<strong>of</strong>iles andthose obtained experimentally showed a good agreement.These authors also analyzed the effect <strong>of</strong> the attrition factorunder the influence <strong>of</strong> the flow condition and the eccentricity<strong>of</strong> the arrangement.M. Carrasco Teja et al [6]: (2009) analyzed the effects <strong>of</strong>rotation and axial motion <strong>of</strong> the inner cylinder <strong>of</strong> an eccentricannular duct during the displacement flow between twoNewtonian fluids <strong>of</strong> differing density and viscosity. Theannulus is assumed narrow and oriented near the horizontal.The main application is the primary cementing <strong>of</strong> horizontaloil and gas wells, in which casing rotation and reciprocation isbecoming common. In this application it is usual for thedisplacing fluid to have a larger viscosity than the displacedfluid. They show that steady traveling wave displacementsmay occur, as for the situation with stationary walls. For smallbuoyancy numbers and when the annulus is near to concentric,the interface is nearly flat and a perturbation solution can befound analytically. This solution shows that rotation reducesthe extension <strong>of</strong> the interface in the axial direction and alsoresults in an azimuthal phase shift <strong>of</strong> the steady shape awayfrom a symmetrical pr<strong>of</strong>ile. They said that the phase shiftresults in the positioning <strong>of</strong> heavy fluid over light fluid alongsegments <strong>of</strong> the interface. When the axial extension <strong>of</strong> theinterface is sufficiently large, this leads to a local buoyancydrivenfingering instability, for which a simple predictivetheory is advanced. Over longer times, the local fingering isreplaced by steady propagation <strong>of</strong> a diffuse interfacial regionthat spreads slowly due to dispersion. Slow axial motion <strong>of</strong> theannulus walls on its own is apparently less interesting. Thereis no breaking <strong>of</strong> the symmetry <strong>of</strong> the interface and hence noinstability.Nouri et al [7]: (1993) who experimentally evaluated theannular flow <strong>of</strong> Newtonian and non-Newtonian fluids insituations above the laminar flow, using the laser anemometrytechnique to quantify the axial, radial and tangential velocitypr<strong>of</strong>iles. Their experimental determinations, however, did notpredict the effect <strong>of</strong> internal axis rotation.Md Mamunur Rashid [8]: (2008) studied a detailedcomputational investigation on the Newtonian fluid flowthrough concentric annuli with centre body rotation withglucose as the working fluid will be carried out. He confinedflow through concentric annuli with centre body rotation isexamined numerically by solving the modified Navier-Stokesequations. He measured the axial and tangential components<strong>of</strong> velocity is presented in non-dimensional form fora Newtonian fluid. The annular geometry consists <strong>of</strong> a rotatingCentre body with angular speed <strong>of</strong> 126 rpm and a radius ratio<strong>of</strong> 0.506. He integrated continuity and the momentumequations numerically with the aid <strong>of</strong> a finite – volumemethod. Your numerical predictions have been confirmed bycomparing them with experimentally derived axial andtangential velocity pr<strong>of</strong>iles obtained for a Newtonian. For theNewtonian (Glucose) fluid, he was carried out for Reynold’snumber <strong>of</strong> 800 and 1200.II. GOVERNING EQUATIONS AND BOUNDARYCONDITIONSConsider steady state, isothermal, laminar, and fullydeveloped flow <strong>of</strong> fluids for which the density and theviscosity are constant. The governing partial differentialequation in cylindrical coordinate for this case become [12]Continuity equation:May 2013ATE-80302020©Asian-Transactions40

Asian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 02inner cylinder with rotation, the pr<strong>of</strong>ile <strong>of</strong> axial velocity willdraft away from ξ = 0.5 due to the rotation will generate acomponent will added to the axial velocity. This effect is verysmall at sector (C). When rotation began increasing <strong>of</strong> innercylinder in sectors B & C axial velocity increases withincreasing the angular speed <strong>of</strong> inner cylinder.While sector (A) related inversely between u and ω.Finally sector (D) have a different behavior compared with theother cases where ω = (150 – 250) rpm with increasing <strong>of</strong>axial velocity.Finally Figs. (13 & 18) shows increasing the pressuredrop with angular speed increase in both cases i.e., ε = 0.2 &0.4, note increasing in pressure drop at ε = 0.2 greater than it'sat ε = 0.4.The nature <strong>of</strong> the increase in the axial pressuregradient is clearly a function <strong>of</strong> the geometry <strong>of</strong> the annulus.The effect <strong>of</strong> inertia on the axial pressure gradient,briefly, the effect <strong>of</strong> inertia is to increase the magnitude <strong>of</strong> theaxial pressure gradient, and the size <strong>of</strong> this increase is afunction <strong>of</strong> inner-cylinder rotation speed and the geometry <strong>of</strong>the annulus in question. Ultimately, these inertial effects mustarise from the coupling <strong>of</strong> the equations in the Navier–Stokesset as the flow is three-dimensional [9].Thus two graphs are shown that increasing <strong>of</strong> angularspeed <strong>of</strong> inner cylinder lead to increase in pressure drop ontwo extreme edge <strong>of</strong> pipe.VI. CONCLUSIONS1. The results gave good agreement with other publishedresearchers.2. Axial velocity at sector (A) increase with increasing <strong>of</strong>angular speed <strong>of</strong> inner cylinder at ε = 0.2 and 0.4.3. Axial velocity decrease with increasing <strong>of</strong> angular speed <strong>of</strong>inner cylinder in sector (C) at ε = 0.2, also in sectors (B, C) atε = 0.4.4. When inner cylinder start to rotate axial velocity pr<strong>of</strong>ilebecome non-symmetrical about centerline radial gap <strong>of</strong>passage.5. Pressure gradient increase with increasing <strong>of</strong> angular speedin both case i.e., ε = 0.2, 0.4 but at ε = 0.2 greater than at ε =0.4 along range <strong>of</strong> angular speed i.e., ω = (0 to 250) rpm.Process Systems Engineering, ENPROMER, Costa Veroe-Rj-Brazil,2003.[4] Escudier, M.P., Gouldson, I.W., Oliveira, P.J., Pinho, F.T., "<strong>Effect</strong>s <strong>of</strong>inner cylinder rotation on laminar flow <strong>of</strong> a Newtonian fluid through aneccentric annulus", Int. J. Heat Fluid Flow, Vol. 21, p. 92–103, 2000.[5] Escudier, M. P.; Oliveira, P. J.; Pinho, F. T.; Simth, S., "Fully developedlaminar flow <strong>of</strong> purely viscous non-Newtonian liquids annuli, Includingthe effects <strong>of</strong> eccentricity and inner – cylinder rotation", InternationalJournal <strong>of</strong> Heat and Fluid Flow, Vol. 33, p. 52-73, 2002.[6] M. Carrasco Teja & I. A. Frigaard, "Displacement flows in horizontal,narrow, eccentric annuli with a moving inner cylinder", Physic <strong>of</strong> Fluids,Vol. 21, 073102, 2009.[7] J.M. Nouri, H. Umur; J.H. Whitelaw, "Flow <strong>of</strong> Newtonian and non-Newtonian fluids in concentric and eccentric annuli", J. Fluid Mech.,Vol. 253, pp. 617-641, 1993.[8] Md Mamunur Rashid, "CFD for Newtonian Glucose fluid flow throughconcentric annuli with centre body rotation", International Journal onScience and Technology (IJSAT) Vol. II, Issue V, pp. 180 – 186, 2008.[9] S.Wan, D. Morrison, and I.G. Bryden, "The Flow <strong>of</strong> Newtonian AndInelastic Non-Newtonian Fluids in Eccentric Annuli With <strong>Inner</strong>-<strong>Cylinder</strong> Rotation", Theoret. Comput. Fluid Dynamics, Vol. 13, pp.349–359, 2000.RECOMMENDATIONSAn extension <strong>of</strong> the present work is recommended fora future works includes:1. Investigation <strong>of</strong> the unsteady flow and study the vortexshedding process.2. Study effect <strong>of</strong> rotation <strong>of</strong> outer cylinder as well asrotation <strong>of</strong> inner cylinder.3. Study effect <strong>of</strong> entrance length that happened beforefully developed flow region.4. Insert effect <strong>of</strong> gravity <strong>of</strong> fluid on the pressure andvelocity pr<strong>of</strong>ile.5. Take in consider the inner cylinder rotates at highrotational speed (turbulence model).REFERENCES[1] C. Shu, L. Wang, Y. T. Chew, N. Zhao, "Numerical Study <strong>of</strong> EccentricCouette – Taylor Flows and <strong>Effect</strong> <strong>of</strong> Eccentricity on Flow Patterns",Theoretical Computational Fluid Dynamics, Vol. 18, pp. 43 – 59, 2004.[2] Hua-Shu Dou, Boo Cheong Khoo, Khoon Seng Yeo, "Energy lossdistribution in the plane Couette flow and the Taylor–Couette flowbetween concentric rotating cylinders", International Journal <strong>of</strong> ThermalSciences Vol. 46, pp. 262–275, 2007.[3] C. H. Ataíde, F. A. R. Pereira e M. A. S. Barrozo, "CFD Predictions OfDrilling Fluids <strong>Velocity</strong> Pr<strong>of</strong>iles In Horizontal Annular Flow", 2 ndMercosur Congress on Chemical Engineering, 4 th Mercosur Congress onMay 2013ATE-80302020©Asian-Transactions42

uAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 02Fig. 1. Grid generation <strong>of</strong> ε = 0.2, κ = 0.5Bωr, ϕCR IeϕAR OFig. 2. Annular section in two dimensions with cylindrical coordinateD21.5ACA[4]C [4]10.500 0.2 0.4 z 0.6 0.8 1Fig. 3. Comparison between present study and Ref. [4] <strong>of</strong> non-dimensional axial velocity pr<strong>of</strong>ile for Re = 105, ε = 0.2 at sectors A and CMay 2013ATE-80302020©Asian-Transactions43

vvuAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 0221.5B [4]D [4]BD10.500 0.2 0.4 z 0.6 0.8 1Fig. 4. Comparison between present study and Ref. [4] <strong>of</strong> non-dimensional axial velocity pr<strong>of</strong>ile for Re = 105, ε = 0.2 at sectors B and D10.80.6A [4]C [4]AC0.40.200 0.2 0.4 z 0.6 0.8 1Fig. 5. Comparison between present study and Ref. [4] <strong>of</strong> non-dimensional tangential velocity pr<strong>of</strong>ile for Re = 105, ε = 0.2 at sectors A and C10.80.6B [4]D [4]BD0.40.200 0.2 0.4 z 0.6 0.8 1May 2013ATE-80302020©Asian-Transactions44

uvuAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 02Fig. 6. Comparison between present study and Ref. [4] <strong>of</strong> non-dimensional tangential velocity pr<strong>of</strong>ile for Re = 105, ε = 0.2 at sectors B and D2.521.5A [4]C [4]AC10.500 0.2 0.4 0.6 0.8 1zFig. 7. Comparison between present study and Ref. [4] <strong>of</strong> non-dimensional axial velocity pr<strong>of</strong>ile for Re = 105, ε = 0.5 at sectors A and C10.80.6A [4]C [4]AC0.40.20-0.20 0.2 0.4 0.6 0.8 1zFig. 8. Comparison between present study and Ref. [4] <strong>of</strong> non-dimensional tangential velocity pr<strong>of</strong>ile for Re = 105, ε = 0.5 at sectors A and C21.510.500 0.2 0.4 z 0.6 0.8 1Fig. 9. Non-dimensional axial velocity <strong>of</strong> sector (A) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.2 and κ = 0.5May 2013ATE-80302020©Asian-Transactions45

uuuAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 021.81.61.41.210.80.60.40.200 0.2 0.4 z 0.6 0.8 1Fig. 10. Non-dimensional axial velocity <strong>of</strong> sector (B) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.2 and κ = 0.51.41.210.80.60.40.200 0.2 0.4 z 0.6 0.8 1Fig. 11. Non-dimensional axial velocity <strong>of</strong> sector (C) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.2 and κ = 0.51.61.41.210.80.60.40.200 0.2 0.4 z 0.6 0.8 1May 2013ATE-80302020©Asian-Transactions46

uupAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 02Fig. 12. Non-dimensional axial velocity <strong>of</strong> sector (D) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.2 and κ = 0.5240023502300p225022000 50 100 150 200 250Fig. 13. Axial pressure gradient (Pa/m) vs. inner cylinder rotation (rpm) at ε = 0.2, κ = 0.52.521.510.500 0.2 0.4 0.6 0.8 1zFig. 14. Non-dimensional axial velocity <strong>of</strong> sector (A) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.4 and κ = 0.51.81.61.41.210.80.60.40.200 0.2 0.4 0.6 0.8 1zFig. 15. Non-dimensional axial velocity <strong>of</strong> sector (B) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.4 and κ = 0.5May 2013ATE-80302020©Asian-Transactions47

puuAsian Transactions on Engineering (ATE ISSN: 2221-4267) Volume 03 Issue 021.210.80.60.40.200 0.2 0.4 z 0.6 0.8 1Fig. 16. Non-dimensional axial velocity <strong>of</strong> sector (C) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.4 and κ = 0.51.210.80.60.40.200 0.2 0.4 z 0.6 0.8 1Fig. 17. Non-dimensional axial velocity <strong>of</strong> sector (D) vs. non-dimensional distance from wall <strong>of</strong> inner to outer cylinder (radial gap) at ε = 0.4 and κ = 0.524002300220021002000190018000 50 100 150 200 250Fig. 18. Axial pressure gradient (Pa/m) vs. inner cylinder rotation (rpm) at ε = 0.4, κ = 0.pMay 2013ATE-80302020©Asian-Transactions48