Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse Effect of Anglar Velocity of Inner Cylinder on Laminar Flow throgh Eccentric Annlar Cross Section Pipe Ressan Faris Hamd * Department of Mechanical Engineering, Kfa University, Iraq Corresponding athor; E-mail address: Ressan@yahoo.com Abstract-- This is stdy investigated the effect increasing the anglar velocity on the axial velocity profile and pressre gradient for steady state, three dimensional, incompressible, isothermal, and laminar flow of a Newtonian flid throgh eccentric annlar dct. The reslts are presented for anglar velocity range (, 5,, 5,, and 5) rpm at axial Reynolds nmber based on blk axial velocity with radis ratio.5 where FLUENT software version (6.) is sed to solve continity and momentm eqations. The reslts of axial and tangential velocity for for sectors were fond to be in good agreement with other previosly pblished research at eccentricity are. and.5. The reslts show that the axial velocity varies with varying in rotational speed of inner cylinder and the axial pressre gradient increases with increasing in rotational speed. Also when inner cylinder is rotated, the axial velocity profile become nonaxisymmetry arond center of radial gap of annlar pipe at eccentricity of. and.4. Index Term-- annlar, rotate cylinder, laminar, eccentric NOMENCLATURE Latin Description symbols displacement of inner cylinder axis from e oter cylinder axis (m) radial distance from axis of inner cylinder r (m) p' Axial pressre gradient (Pa/m) R I oter radis of inner cylinder (m) R O inner radis of oter cylinder (m) Re axial Reynolds nmber ρuδ/μ (--) non-dimensional axial component of velocity (--) non-dimensional tangential component of v velocity (--) axial component of velocity (m/s) v tangential component of velocity (m/s) U blk axial velocity (m/s) non-dimensional radial component of w velocity (m/s) Cartesian coordinate in horizontal direction x (m) Non-dimensional vale of distance from y oter wall (m) Greek symbols ξ Description Non-dimensional distance from wall of inner cylinder (--) κ Radis ratio R I /R O (--) ε Eccentricity e/δ (--) ρ density of the air (kg/m 3 ) Azimthal location with respect to inner ϕ cylinder (degree) δ Annlar gap width R O - R I (m) ω anglar velocity of inner cylinder (rpm) µ dynamic viscosity of the flid (kg/m.s) v kinematic viscosity of the flid (m /s) symbols Abbreviations CFD Comptational Flid Dynamic I. INTRODUCTION The classical Coette flow problem consists of infinitely long concentric cylinders and an incompressible Newtonian flid between them. For a system with a rotating inner cylinder and a stationary oter cylinder, the flid flow will pass on stable circlar Coette flow and steady axisymmetric Taylor vortex flow for the vale of Taylor nmber Ta (or Reynolds nmber Re) less than and greater than a critical vale Ta c (or Re c ) respectively []. The flow device of Taylor Coette flow comprising concentric rotating cylinders is widely sed in many indstrial and research processes fond in chemical, mechanical and nclear engineering. The device can be only one cylinder rotating and the other at rest, or two cylinders rotating in the same or conter directions. The accrate calclation of the flow property is important even from the standpoint of the normal operation of the device. The distribtion of energy loss in the device may greatly inflence the indstrial process of mixing, diffsion, heat transfer, and flow stabilities, etc. []. C. H. Ataide et al [3]: (3) analyzed the flid flow in annlar regions is an area of great interest in the petrolem indstry, both in the drilling and in the artificial rising of the petrolem. Throgh the nmerical simlation, sing the compter flid dynamics commercial code (CFD). They investigated flow of non-newtonian flids flow throgh the annlar space formed by two tbes in concentric and eccentric May 3 ATE-83 Asian-Transactions 39
Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse arrangements of a horizontal system. They stdied contemplates the prediction of effects of viscosity, eccentricity, flow and inner axis rotation velocity over the tangential and axial velocity profiles and over the hydrodynamic losses, considering that these two variables are relevant in the nderstanding of the drilling flids flow. They evalated the performance by sing nmerical techniqe, and compared the obtained reslts with other reported works. C. Sh et al []: (4) sed the differential qadratre (DQ) method to simlate the eccentric Coette Taylor vortex flow in an annls between two eccentric cylinders with rotating inner cylinder and stationary oter cylinder. Their approach combining the SIMPLE (semi-implicit method for pressre linked eqations). They proposed to solve the time-dependent, three-dimensional incompressible Navier Stokes eqations in the primitive variable form. They obtained eccentric steady Coette Taylor flow patterns from the soltion of threedimensional Navier Stokes eqations. For steady eccentric Taylor vortex flow, detailed flow patterns were obtained and analyzed. The effect of eccentricity on the eccentric Taylor vortex flow pattern was also stdied. Escdier et al [4]: () showed that there is excellent agreement between their calclations for a Newtonian flid and experimental data for annli with %, 5%, and 8% eccentricity. Their reslts are reported for calclations of the flow field, wall shear stress distribtion and friction factor for a range of vales of eccentricity, radis ratio, and Taylor nmber. They showed the axial component of velocity is directly affected by the radial/tangential velocity field and rotation of the inner cylinder is fond to have a strong inflence on the axial velocity distribtion. As the Taylor nmber is increased the friction factor for high vales of ε >.9 increases. Escdier et al. [5]: () evalated the effect of internal cylinder rotation on the laminar flow of non-newtonian flids in an eccentric annlar region, compared their nmerical simlations of two-dimensional flow against experimental data. Their comparison of simlated velocity profiles and those obtained experimentally showed a good agreement. These athors also analyzed the effect of the attrition factor nder the inflence of the flow condition and the eccentricity of the arrangement. M. Carrasco Teja et al [6]: (9) analyzed the effects of rotation and axial motion of the inner cylinder of an eccentric annlar dct dring the displacement flow between two Newtonian flids of differing density and viscosity. The annls is assmed narrow and oriented near the horizontal. The main application is the primary cementing of horizontal oil and gas wells, in which casing rotation and reciprocation is becoming common. In this application it is sal for the displacing flid to have a larger viscosity than the displaced flid. They show that steady traveling wave displacements may occr, as for the sitation with stationary walls. For small boyancy nmbers and when the annls is near to concentric, the interface is nearly flat and a pertrbation soltion can be fond analytically. This soltion shows that rotation redces the extension of the interface in the axial direction and also reslts in an azimthal phase shift of the steady shape away from a symmetrical profile. They said that the phase shift reslts in the positioning of heavy flid over light flid along segments of the interface. When the axial extension of the interface is sfficiently large, this leads to a local boyancydriven fingering instability, for which a simple predictive theory is advanced. Over longer times, the local fingering is replaced by steady propagation of a diffse interfacial region that spreads slowly de to dispersion. Slow axial motion of the annls walls on its own is apparently less interesting. There is no breaking of the symmetry of the interface and hence no instability. Nori et al [7]: (993) who experimentally evalated the annlar flow of Newtonian and non-newtonian flids in sitations above the laminar flow, sing the laser anemometry techniqe to qantify the axial, radial and tangential velocity profiles. Their experimental determinations, however, did not predict the effect of internal axis rotation. Md Mamnr Rashid [8]: (8) stdied a detailed comptational investigation on the Newtonian flid flow throgh concentric annli with centre body rotation with glcose as the working flid will be carried ot. He confined flow throgh concentric annli with centre body rotation is examined nmerically by solving the modified Navier-Stokes eqations. He measred the axial and tangential components of velocity is presented in non-dimensional form for a Newtonian flid. The annlar geometry consists of a rotating Centre body with anglar speed of 6 rpm and a radis ratio of.56. He integrated continity and the momentm eqations nmerically with the aid of a finite volme method. Yor nmerical predictions have been confirmed by comparing them with experimentally derived axial and tangential velocity profiles obtained for a Newtonian. For the Newtonian (Glcose) flid, he was carried ot for Reynold s nmber of 8 and. II. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS Consider steady state, isothermal, laminar, and flly developed flow of flids for which the density and the viscosity are constant. The governing partial differential eqation in cylindrical coordinate for this case become [] Continity eqation: May 3 ATE-83 Asian-Transactions 4
Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse v wr r -Momentm eqation: v p w r () r r z r r r r v-momentm: v v v wv p 3 v w w r r r r r r r r r r v w r r r w-momentm: w v w v p w w r r r r r r r r v w r r r with bondary conditions: = v = w = on the oter cylinder; = w = and v = ω.r I on the inner cylinder; m c at inlet (axial flow in z-direction) other terms define as: R R R O R I I O e R O R I v v R i U r Ri ζ R R o U Re Ta i R I 3 III. GRID GENERATION Total grid for the problem is generated by sing commercial grid generator GAMBIT (..3) in three dimension of annlar passage at eccentricity of. &.4. Grid strctre is created here and it consists of non-niform hexahedral wedge element having 9, cells and,96 nodes. The grid was employed of the problem consists of 4 radial, circmferential and 4 axial grid-lines as shown in Fig.. () (3) (4) IV. VALIDATION The reslts of velocity profile for flow are presented with help of cylindrical coordinates, where the x axis of the system is dimensionless axial flow axis and radial direction axis along the gap of annlar pipe of for sectors (A, B, C, and D) as shown in Fig. (). The calclated reslts for ε =.,.5 for rotation of inner cylinder Re = 5 and Ta = 3 are in good agreement with the velocity profile data of reference [4]. The mass flow rate sed to determine the axial velocity vale. Figs. (3 to 8) were represented profile of axial and tangential velocity components with dimensionless are plotted as a fnction of the dimensionless position gave agreement between the reslts of FLUENT (6.) and reference [4] of the sectors (A, B, C, and D) at ε =. also the sectors (A and C) at ε =.5. V. RESULTS Nmerical reslts are achieved by FLUENT (6.) to solve continity and momentm (Navier-Stokes eqations) in 3D-steady state, cylindrical coordinates. The reslts are presented in Figs. (9,,, and ) which is represented the relationship between non-dimensional distance from wall of inner cylinder toward radial to oter diameter (ξ) and nondimensional axial velocity () of for sectors (A, B, C, and D) varying with the vale of anglar speed of inner cylinder for ranges (, 5,, 5,, and 5) rpm at ε =.. While Figs. (4, 5, 6, and 7) similarly to the pervios figres except eccentricity is eqal to.4. Figs. (3 & 8) are represented pressre drop or gradient (Pa/m) with anglar speed of inner cylinder (rpm) at ε =. &.4 respectively. For all cases at ω > rpm max. axial velocity will be not at center de to rotation affect on profile of axial component. Therefore add drift on center location. Figs. (9 to ) for that have eccentricity of. shows effect increases of anglar speed of inner cylinder on axial velocity in relative to sector (A), note that axial velocity is decrease with the increasing in rotation speed gradally, while sector (C) this velocity increases when rotation speed is increasing, bt sector (B) no rotation inner cylinder the vales of axial velocity will be smaller than in case rotation after ω increase from to 5 rpm lead to vales of axial velocity is decrease above nonrotation vales. Bt for sector (D) the vales of axial velocity will decreases when inner cylinder is rotate also after ω increase from 5 to 5 rpm will lead to axial velocity vales increases along the annlar gap. Figs. (4 to 7) represents will the effect of inner cylinder rotation on profile of axial velocity along radial direction of annlar passage in for region sectors (A, B, C, and D) that have eccentricity of.4, for all cases each crve at ω = is symmetrical and max. Vale at ξ =.5 bt when start May 3 ATE-83 Asian-Transactions 4
Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse inner cylinder with rotation, the profile of axial velocity will draft away from ξ =.5 de to the rotation will generate a component will added to the axial velocity. This effect is very small at sector (C). When rotation began increasing of inner cylinder in sectors B & C axial velocity increases with increasing the anglar speed of inner cylinder. While sector (A) related inversely between and ω. Finally sector (D) have a different behavior compared with the other cases where ω = (5 5) rpm with increasing of axial velocity. Finally Figs. (3 & 8) shows increasing the pressre drop with anglar speed increase in both cases i.e., ε =. &.4, note increasing in pressre drop at ε =. greater than it's at ε =.4. The natre of the increase in the axial pressre gradient is clearly a fnction of the geometry of the annls. The effect of inertia on the axial pressre gradient, briefly, the effect of inertia is to increase the magnitde of the axial pressre gradient, and the size of this increase is a fnction of inner-cylinder rotation speed and the geometry of the annls in qestion. Ultimately, these inertial effects mst arise from the copling of the eqations in the Navier Stokes set as the flow is three-dimensional [9]. Ths two graphs are shown that increasing of anglar speed of inner cylinder lead to increase in pressre drop on two extreme edge of pipe. VI. CONCLUSIONS. The reslts gave good agreement with other pblished researchers.. Axial velocity at sector (A) increase with increasing of anglar speed of inner cylinder at ε =. and.4. 3. Axial velocity decrease with increasing of anglar speed of inner cylinder in sector (C) at ε =., also in sectors (B, C) at ε =.4. 4. When inner cylinder start to rotate axial velocity profile become non-symmetrical abot centerline radial gap of passage. 5. Pressre gradient increase with increasing of anglar speed in both case i.e., ε =.,.4 bt at ε =. greater than at ε =.4 along range of anglar speed i.e., ω = ( to 5) rpm. Process Systems Engineering, ENPROMER, Costa Veroe-Rj-Brazil, 3. [4] Escdier, M.P., Goldson, I.W., Oliveira, P.J., Pinho, F.T., "Effects of inner cylinder rotation on laminar flow of a Newtonian flid throgh an eccentric annls", Int. J. Heat Flid Flow, Vol., p. 9 3,. [5] Escdier, M. P.; Oliveira, P. J.; Pinho, F. T.; Simth, S., "Flly developed laminar flow of prely viscos non-newtonian liqids annli, Inclding the effects of eccentricity and inner cylinder rotation", International Jornal of Heat and Flid Flow, Vol. 33, p. 5-73,. [6] M. Carrasco Teja & I. A. Frigaard, "Displacement flows in horizontal, narrow, eccentric annli with a moving inner cylinder", Physic of Flids, Vol., 73, 9. [7] J.M. Nori, H. Umr; J.H. Whitelaw, "Flow of Newtonian and non- Newtonian flids in concentric and eccentric annli", J. Flid Mech., Vol. 53, pp. 67-64, 993. [8] Md Mamnr Rashid, "CFD for Newtonian Glcose flid flow throgh concentric annli with centre body rotation", International Jornal on Science and Technology (IJSAT) Vol. II, Isse V, pp. 8 86, 8. [9] S.Wan, D. Morrison, and I.G. Bryden, "The Flow of Newtonian And Inelastic Non-Newtonian Flids in Eccentric Annli With Inner- Cylinder Rotation", Theoret. Compt. Flid Dynamics, Vol. 3, pp. 349 359,. RECOMMENDATIONS An extension of the present work is recommended for a ftre works incldes:. Investigation of the nsteady flow and stdy the vortex shedding process.. Stdy effect of rotation of oter cylinder as well as rotation of inner cylinder. 3. Stdy effect of entrance length that happened before flly developed flow region. 4. Insert effect of gravity of flid on the pressre and velocity profile. 5. Take in consider the inner cylinder rotates at high rotational speed (trblence model). REFERENCES [] C. Sh, L. Wang, Y. T. Chew, N. Zhao, "Nmerical Stdy of Eccentric Coette Taylor Flows and Effect of Eccentricity on Flow Patterns", Theoretical Comptational Flid Dynamics, Vol. 8, pp. 43 59, 4. [] Ha-Sh Do, Boo Cheong Khoo, Khoon Seng Yeo, "Energy loss distribtion in the plane Coette flow and the Taylor Coette flow between concentric rotating cylinders", International Jornal of Thermal Sciences Vol. 46, pp. 6 75, 7. [3] C. H. Ataíde, F. A. R. Pereira e M. A. S. Barrozo, "CFD Predictions Of Drilling Flids Velocity Profiles In Horizontal Annlar Flow", nd Mercosr Congress on Chemical Engineering, 4 th Mercosr Congress on May 3 ATE-83 Asian-Transactions 4
Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse Fig.. Grid generation of ε =., κ =.5 B ω r, ϕ C R I e ϕ A R O Fig.. Annlar section in two dimensions with cylindrical coordinate D.5 A C A[4] C [4].5..4 z.6.8 Fig. 3. Comparison between present stdy and Ref. [4] of non-dimensional axial velocity profile for Re = 5, ε =. at sectors A and C May 3 ATE-83 Asian-Transactions 43
v v Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse.5 B [4] D [4] B D.5..4 z.6.8 Fig. 4. Comparison between present stdy and Ref. [4] of non-dimensional axial velocity profile for Re = 5, ε =. at sectors B and D.8.6 A [4] C [4] A C.4...4 z.6.8 Fig. 5. Comparison between present stdy and Ref. [4] of non-dimensional tangential velocity profile for Re = 5, ε =. at sectors A and C.8.6 B [4] D [4] B D.4...4 z.6.8 May 3 ATE-83 Asian-Transactions 44
v Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse Fig. 6. Comparison between present stdy and Ref. [4] of non-dimensional tangential velocity profile for Re = 5, ε =. at sectors B and D.5.5 A [4] C [4] A C.5..4.6.8 z Fig. 7. Comparison between present stdy and Ref. [4] of non-dimensional axial velocity profile for Re = 5, ε =.5 at sectors A and C.8.6 A [4] C [4] A C.4. -...4.6.8 z Fig. 8. Comparison between present stdy and Ref. [4] of non-dimensional tangential velocity profile for Re = 5, ε =.5 at sectors A and C.5.5..4 z.6.8 Fig. 9. Non-dimensional axial velocity of sector (A) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =. and κ =.5 May 3 ATE-83 Asian-Transactions 45
Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse.8.6.4..8.6.4...4 z.6.8 Fig.. Non-dimensional axial velocity of sector (B) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =. and κ =.5.4..8.6.4...4 z.6.8 Fig.. Non-dimensional axial velocity of sector (C) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =. and κ =.5.6.4..8.6.4...4 z.6.8 May 3 ATE-83 Asian-Transactions 46
p Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse Fig.. Non-dimensional axial velocity of sector (D) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =. and κ =.5 4 35 3 p 5 5 5 5 Fig. 3. Axial pressre gradient (Pa/m) vs. inner cylinder rotation (rpm) at ε =., κ =.5.5.5.5..4.6.8 z Fig. 4. Non-dimensional axial velocity of sector (A) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =.4 and κ =.5.8.6.4..8.6.4...4.6.8 z Fig. 5. Non-dimensional axial velocity of sector (B) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =.4 and κ =.5 May 3 ATE-83 Asian-Transactions 47
p Asian Transactions on Engineering (ATE ISSN: -467) Volme 3 Isse..8.6.4...4 z.6.8 Fig. 6. Non-dimensional axial velocity of sector (C) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =.4 and κ =.5..8.6.4...4 z.6.8 Fig. 7. Non-dimensional axial velocity of sector (D) vs. non-dimensional distance from wall of inner to oter cylinder (radial gap) at ε =.4 and κ =.5 4 3 9 8 5 5 5 Fig. 8. Axial pressre gradient (Pa/m) vs. inner cylinder rotation (rpm) at ε =.4, κ =. p May 3 ATE-83 Asian-Transactions 48