non-newtonian fluid past irregular particles from …

NON-NEWTONIAN FLUID PASTIRREGULAR PARTICLES FROMLOW TO MODERATE REYNOLDS

NUMBERS

N Nanda Kumar1, SSSR Sarathbabu Duvvuri2,1Department of Pharmaceutical Engineering

1B.V. RAJU Institute of Technology Narsapur.2Department of Electrical and Electronics Engineering

2Shri Vishnu Engineering College for Women [email protected]

[email protected]

July 19, 2018

Abstract

The flow of Newtonian and non-Newtonian fluids pastdifferent shaped solid particles has been a subject of inter-est due to their many industrial applications. This studyshowed numerical results for the flow of Newtonian as wellas non-Newtonian fluids (n = 0.8 and 0.6) past ellipsoidalparticles over a range of parameters: eccentricity, 0.5< e>2.5 and Reynolds number, 10<Re>100 obtained by us-ing computational fluid dynamics. The present results arevalidated by comparing them with various numerical dragvalues reported in the literature. It has been seen that forellipsoidal solid particles, the effect of individual and totaldrag coefficients is less affected at small Reynolds numbers,but as increasing the Reynolds number the total drag coeffi-cient decreases due to the decrease in the pressure coefficientand due to reduced effect of viscous forces on the surface of

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International Journal of Pure and Applied MathematicsVolume 120 No. 6 2018, 7929-7948ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

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the solid particle. A significant difference in total drag andviscous has been observed for power law fluids compare toNewtonian fluids. Based on the present numerical results, asimple predictive correlation is proposed which can be usedto estimate the drag coefficient in an unknown system.

Keywords: Ellipsoidal Particles; eccentricity; Reynoldsnumber; Drag coefficient

Nomenclature:CD total drag coefficient (dimensionless)CDF total drag coefficient (dimensionless)CDP total drag coefficient (dimensionless)CP pressure coefficient (dimensionless)a major dimension of ellipseb minor dimension of ellipsee eccentric factorFD drag force on the ellipse particlesP pressure (Pa)Re Reynolds number, ρVD/µ (dimensionless)U Velocity at the inlet (m/s)V average velocity (m/s)Vx X-direction velocity (m/s)Vy Y-direction velocity (m/s)µ viscosity of the fluid (Pa s)τw wall shearρ density (kg/m3)D Pipe diameter (m)f Fanning friction factorg acceleration due to gravity LT -2L length of pipe (m)m power-law consistency coefficientn power-law flow behaviour indexn’ apparent power-law indexN power (W)

1 INTRODUCTION

In many real applications, we need to know the force required tomove a solid object through a surrounding fluid, or conversely, the

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force that a moving fluid exerts on a solid as the fluid flow pastit. Many processes for the separation of particles of various sizes,shape and materials depend on their behavior when subjected tothe action of moving fluid. Frequently, the liquid phase may ex-hibit complex non-Newtonian behavior whose characteristics maybe measured using viscometry. Furthermore, it is often necessary tocalculate the fluid dynamic drag on solid Particles in process equip-ment, for example for slurry pipelines, fixed and fluidized beds.The phenomena of flow separation, wakes formation and drag be-haviour in flow past various types of bluff bodies has acquired greatinterest amongst many researchers because of their fundamentalsignificance in understanding the flow physics and their practicalimportance in hydrodynamic applications. In general, engineeringapplications often involve flows over simple and complex bluff bod-ies like spheres, cylinders, triangles; cubes, diamonds, ellipsoidalparticles and other arbitrarily shaped solid particles. In such flows,parameters such as angle-of-attack greatly influence the nature ofseparation and the wake structure. Therefore, a fundamental studyof flow over complex shapes of ellipsoidal particles would signif-icantly augment the current understanding of the physics of theflow and drag behaviour of flow past bluff bodies. In recent yearsconsiderable effort has been directed at investigating the flow anddrag behavior of various shapes of solid particles (spheres, cylinders,squares, etc.) settling in Newtonian liquids [1] and to some extentin non-Newtonian fluids[2]. An inspection of the recent reviewsof the pertinent literature shows that the sedimentation behaviorof solid spheres, circular and square cylinders in Newtonian andfollowed by those in purely viscous fluids (mostly power-law) hasbeen studied most thoroughly, at least, in the moderate range ofpertinent parameters [3][4][5][6][2]. Based on a combination of thenumerical and experimental studies, reliable results are now avail-able on drag coefficient Reynolds number behavior; flow separationand wake formation behaviour for spheres, circular cylinders andsquare cylinders falling in unconfined fluids up to about Reynoldsnumber of 1000. On the other hand, ellipsoidal particles, which aremore general geometrical configurations than the spherical or cylin-drical particles and provide a richer flow behavior characteristic oftypical engineering flow configurations. For these shapes, changesin aspect ratio ranging from 0.5 ≤ e ≤ 2.5 allow for different shapes

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from disk-like (e < 1) to cylinder like (e > 1) and includes spheresfor which e = 1. But there is lack of systematic study pertainingto the flow and drag phenomenon of ellipsoidal particles in New-tonian liquids. There have been few studies available on flow ofNewtonian liquids past ellipsoidal or spheroidal particles in limit-ing conditions. Analytical expressions for the drag force acting onprolates and oblates are available in creeping flow regime [7]. Mil-itzer et al. [8] proposed a correlation for drag spheroidal particles inNewtonian media on the basis of the available experimental data.The steady and laminar flow past spheroids have been numericallystudied by Masliyah and Epstein [9] and Comer and Kleinstreuer[10]. Their results show that the surface pressure variations anddrag coefficients are a strong function of the aspect ratio. Brenner[11] presented a review on the settling of a non-spherical particle inan infinite Newtonian fluid under creeping flow conditions.

Blaser[12] used the results obtained by Brenner [11] to evaluatethe hydrodynamic forces acting on an ellipsoidal particle immersedin various flow fields, like, constant, simple shear, two dimensionalstraining and axisymmetric straining flow in the creeping flow re-gion. Wen and Jog [13] numerically studied the steady and varyingthermo-physical property, continuum, and laminar thermal plasmaflow over spheroidal particles. They observed that the effects of flowReynolds number, particle shape, surface and far field temperaturesand variable properties are greatly affect the flow and temperaturefields around the particles. They reported drag values of spheroidalparticles as function of particle surface temperature and free streamtemperatures. However, drag results are not presented for the caseof isothermal spheroidal particles. Tripathi and Chhabra [14] calcu-lated the drag force on a spheroid moving in a power-law liquid inthe convective flow regime. Broday et al.[15] theoretically studiedthe motion of non-neutrally buoyant prolate spheroidal particles invertical shear flows to calculate the hydrodynamic forces and mo-ments acting on such inertial and inertia less particles and theirtrajectories. Hsu et al. (2005)[16] have analytically studied theboundary effect on the drag force acting on a rigid non-sphericalparticle settling in a Carreau fluid in the low to moderate rangeof Reynolds numbers. Rajitha et al. [17] experimentally obtainedthe free settling velocity of cylinders and disks falling in quiescentNewtonian and power-law liquids and proposed modification to the

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existing drag expression for spherical particles falling in Newtonianand power-law fluids by introducing a sphericity factor. There havealso been studies on the evaluation of hydrodynamic force on theflow consisting either spherical particles [18] or spheroidal particles[19] [16] for either of the cases of homogenous and non-homogenousporous flows. However, there is no systematic study reported onthe flow and drag phenomena of single ellipsoidal particles flowingin a quiescent Newtonian liquids as well as non-Newtonian fluidsin the moderate range of Reynolds numbers. Thus this work isaimed to fill this gap and provide an independent correlation tocalculate the drag coefficient acting on ellipsoidal particle in a non-Newtonian liquid in the range of parameters 0.5 ≤ e ≤ 2.5 and 10≤ Re ≤ 100.

2 PROBLEM DESCIPTION

The problem to be considered is shown schematically in Figure 1.Which shows the body of two ellipsoidal particles which has twodifferent cases of ellipse, one is a>b and second one is a<b (a=length of minor axis, b= length of major axis). All these bodies in afluid are subjected to a buoyancy force. While flowing, Skin friction(or viscous drag) and form drag (due to pressure distribution) areoccurred due to relative velocity between the fluid and the solidbody. At low velocities, no separation of boundary layer takesplace, but as velocity increased, separation occurs; skin frictioncomes into existence and a gradual decrease in proportion of thetotal fluid dynamic drag on the immersed object. In Newtonianand power law liquids, For constant e=1 (sphere), n=1 and Rep=103 to 105 the drag co-efficient is nearly constant at 0.40 to 0.45,changing only slightly as the point of boundary layer separationslowly shifts toward the nose of the sphere. Though no analogousinformation is available for the different shapes of solid bodies inpower-law continuous phase, the scant experimental results seemto suggest that the shape factor (e) causes much greater distortionsin flow separation and recirculation at rear of the body. In thispresent study, assumed the fluid is Newtonian and non-Newtonian,so viscosity term has calculate based on power law equation andaims to analyzes the properties of fluid by taking the different values

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of , n = 0.6,0.8,1 , Reynolds number as Re = 10,20,50, 100 and e= b/a (0.5,0.25,1,1.25, and 1.5).

A Cartesian coordinate system (x,y,z) with its origin fixed at thecentre of the solid is employed here with the axis x being directedalong the flow direction as shown in Figure 1(b). Due to the sym-metry, Uz = 0 and no flow variable depends upon the z-coordinate.The flow and mass transfer characteristics in the continuous aregoverned by the equations of continuity, momentum and speciescontinuity.

Figure 1: Schematic representation of the flow past (a) a prolateand (b) an oblate

3 NUMERICAL DETAILS

There are, however, practical reasons why separating flows overcomplex shapes do not lend themselves easily to analytical, exper-imental, or numerical treatment. Due to the complicated nature ofthe flow, theoretical analysis is typically limited to either flow atvery low Reynolds number or flows at early times after an impulsive

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start. Experimental techniques have become very sophisticated inrecent years but an extensive spatial and temporal analysis of thetwo-dimensional (2D) and three-dimensional (3D) spatial and tem-poral analysis of the two-dimensional (2D) and three-dimensional(3D) flow field would quickly overwhelm the available resources.Numerical simulations provide a promising approach to analyzingthis problem. However, there remain a number of issues that needto be addressed, namely intelligent grid generation, efficient solu-tion of the governing equations, and the ability to interpret thegigabytes of data that would be generated from the simulations.Here we consider elliptic particles which are more general geomet-rical configurations than the sphere particles and provide a richerflow behavior characteristic of typical engineering flow configura-tions. For these shapes, changes in eccentricity allow for shapesranging from 0.5≤e≤2.5.

4 COMPUTATIONAL DOMAIN AND

BOUNDRY CONDITIONS

Fig.4 illustrates a schematic representation of the computation mesharound solid particles with eccentricities (e=1, e=0.5, e=2.5) whichare surrounded by an imaginary finite fluid domain. Each area ofparticle is depends on the eccentricity factor ranges from 0.5≤e≤2.5,irrespective of changes in value of major axis b. In our computa-tions it has been assumed that the minor axis length (a= 0.5) andvelocities in y-direction are constant. The relative motion has beenconsidered where the particles remain stationary. The isothermalwater transport process in the computational domain is modeledas a 2D single phase viscous laminar flow. The uniform inlet veloc-ity (calculated by using Reynolds number) is applied as the inletboundary condition. At the outlet the boundary condition wasassigned as pressure outlet and no-slip boundary condition was as-signed on the surface of the solid particles. To simulate the behaviorof total drag coefficient of various particles, distribution of stream-lines along the surface and wakes formation at rear of the solidparticles is carefully setup and details are given below.

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5 COMPUTATIONAL METHODOLOGY

The numerical simulations of the 2D, steady state, laminar flow,segregated, single phase flow in the computation domain was per-formed using ANSYS FLUENT. According to literature Newtonianfluids are consider under time independent fluids, so thereby thenumerical setup supporting steady state laminar flow assumption.No energy equations were considered therefore the conservation ofmass and momentum were the governing equations for the model.

The equations of continuity and momentum for an incompress-ible fluid reduce to:

(1)

x-component:

(2)

y-component:

(3)

Once the fully converged velocity and pressure fields are known,the individual and the total drag coefficients are obtained fromforce option in fluent and similarly streamlines, vorticity contoursand pressure coefficient are obtained from contours option. Theindividual and the total drag coefficients are evaluated directly fromsoftware, which calculated based on the total drag coefficient asdescribed below.

(4)

Where FD is the stress on the surface of the solidThe above formula represents total drag coefficient for sphere.

Here the area taken as area of circle due to the projection areafor sphere was circle in flow direction. But for ellipsoidal particles

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the projected area is taken as area of ellipse. So ellipse area iscalculated based on formula given below:

(5)

Here a = major dimension of the flat oval duct (m, in),b = minor dimension of the flat oval duct (m, in)The equivalent diameter of an ellipse must be required for calcu-

lating of Reynolds numbers, so the equivalent diameter of an ellipsecan be calculated as (Heyt Diaz)

(6)

Where A = cross-sectional area of ellipseP = perimeter of ellipseThe perimeter of an oval duct can be expressed as

(7)

The pressure component Cdp is evaluated as

(8)

And the frictional component Cdf is evaluated as

(9)

The total drag coefficient CD is simply the sum of Cdp and Cdf .The splitting of the total drag into components is instructive notonly in delineating the individual contributions of the shearing (Cdf

) and the normal (Cdp) forces, but it also facilitates the interpre-tation of the surface pressure profiles. Thus, for instance, one canexpect a larger value of Cdp when the pressure recovery is poor inthe rear of the drop than that when the pressure recovery is good.

In the current study, the Reynolds number, is defined based onthe diameter of the particle in the direction normal to the flow, i.e.,on the basis of ’b’.

(10)

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In order to obtain the steady state numerical velocity and pres-sure fields in the entire computational domain, Eqs. (1) - (3) havebeen solved using finite volume method based commercial CFDsoftware ANSYS FLUENT. The convective terms of the momen-tum equation have been descritized using QUICK scheme and thetemporal and viscous terms have been descritized using a centraldifferencing scheme. The inlet velocity and outlet pressure bound-ary conditions are used at inlet and outlet boundaries.

6 VALIDATION OF GRID INDEPEN-

DENCY

Figure 2: computational mesh around e=1, e=2.5, e=0.5

The computational domain consists of 120x400 bi-quadratic el-ements which correspond to 50,526 grid points. The bi-quadraticelements with lesser number of nodes smoothly capture the non-linear variations of the field variables which are in contrast withfinite difference/finite volume solutions available in the data base.In order to assess the accuracy of the numerical procedure, the algo-rithm based on the grid size (120x400) for a sphere enclosure witha fluid domain were compared. There were 400x120 cells meshedin the computation domain as shown in fig on the x-y co-ordinate.As the value of the Reynolds number progressively increases, theboundary layer on the surface of the particle becomes thin. There-fore, a very fine grid (extending beyond the theoretical boundarylayer thickness) is required on surface so that to meet the require-ment of fine grids, we have taken transformation of the grid iny-direction which provides fine grid near the surface, which cap-tures the boundary layer thickness with adequate accuracy. Thisalso provides a somewhat coarser mesh size away from the ellipse

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particles surface where the gradients are less steep, thereby afford-ing a computational economy without sacrificing the accuracy ofthe results.

Having fixed an optimum domain, it is also necessary to choosean optimal grid size which represents a trade-off between the ac-curacy of the results and the computational requirements. Gridcan be a sensitive factor for large Reynolds number flows due toincreasing contribution of convection terms than the viscous terms,thus, grid independence study is to be carried out for large valueof Reynolds number. Table.1 shown a summary of the grid inde-pendence study for two extreme values of eccentricity i.e., e-0.5 ande=2.5 at %Re=200. Here too, any combination of any two gridsproduces results within < 1% however, for the safer side a gridof size 120*400 has been chosen as the optimum grid for the flowstudy. Thus now it is safe to say that the present results are freefrom numerical artifacts such as grid and domain independence.

7 RANGE OF PARAMETERS

The cross-sectional shape of the ellipse has predominant effect onfluid flow characteristics. Thus, a parametric study using CFDhas been carried out with commonly used shapes elliptical particlesat ranges of 0.5≤e≤2.5. The various parameters studied and theranges used in the investigations are as under:

• Effect of Reynolds number at ranges of 1≤Re≤200, has beenstudied for all the shapes of ellipsoidal particles for a fixed

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fluid domain.

• For sphere shape (e=1), a comparative analysis has been car-ried out for total Drag coefficient at the surface of the particle(perpendicular to the flow).

• For elliptical shape, the ratio of major to minor axis, calledeccentric factor (slenderness ratio), has been varied keepingminor axis (perpendicular to flow) constant. The maximumvalue of the eccentric factor (slenderness ratio) analyzed.

Effect of eccentric factor and permanent pressure loss factorhas been investigated for sphere and ellipsoidal shapes. Dimen-sions of the ellipse cross-section have been varied by keeping fixedfluid domain of constant diameter to analyze the wakes and flow ofseparation at various Reynolds numbers.

8 DOMAIN INDEPENDENCE

The size of the domain is more critical at low values of the Reynoldsnumber due to stronger viscous forces compared to convective forcesTable.2 summarize the findings of the domain independence studyfor two extreme values of eccentricity i.e., e=0.5 and e=2.5 at Re=1using three values of the domain sizes. Re=50, 150,250. From thistable, it is clear that almost all domain sizes produce results within<1% for both values of eccentricity factor. Therefore, a domain ofsize R =150 has been considered as an optimum domain.

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9 RESULTS & DISCUSSION

In order to investigate flow behavior of fluid over the surface ofparticle, six different cases of ellipsoidal particles were simulated.Dimensional considerations suggest this problem to be governed bydimensionless groups; the drag coefficient (cd), the Reynolds num-ber (Re), eccentric factor (e) and the power-law index (n). Thiswork, extensive numerical results have been obtained over the fol-lowing ranges of parameters: Re=1, 10, 20, 50, 100, 200: e=0.5,0.75, 1, 1.5, 2, 2.5: n=1(Newtonian fluids). In this work power-lawindex, n, was kept a constant value of one, so it is known to the influ-ence of power-law index on the property of viscosity was negligible.The effect of parameters Re, e, and n on Cd (Drag coefficient), Cdf

(due to friction drag),Cdp (due to pressure drag), surface pressurecoefficient and vorticity magnitude on the surface of the body wasexamined below by varying geometry of solid particles for variousvalues of Reynolds numbers. Before presenting the new results itis desirable to ascertain the accuracy and reliability of the presentresults by comparing them with the literature values. Table.3 shows a summary of comparison of the present values of drag coefficientof solid spheres (e=1) in Newtonian liquid at Re=1, 20, 300 withthose of LeClai et al [20], Graham and Jones [21], Saboni et al [22]and Dhole et al [23]. It is clear from this table that the presentresults are seen to be within 2-3% of the literature values. Further-more, differences of this magnitude are not at all uncommon in suchnumerical studies due to the differences stemming from grid errors,solution procedure,[24] etc.. In view of these factors and based onour comparisons, the present results are believes to be reliable andaccurate to within 2-4%.

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9.1 Drag phenomena for Newtonian Fluids

The above fig. 3 shows the effect of eccentricity on individualand total drag coefficients and on the pressure to friction drag ratio.Regardless of the value of eccentricity factor, both individual andtotal drag coefficients decreases as the value of Reynolds numberincreases.Effect of ’e’ on Cd for fixed power-law index

Fig: 3(a) shows the effect of eccentric factor on the dependenceof the total drag coefficient as a function of the Reynolds numberand the fixed value of power law index n=1. For fixed value ofthe ’n’ and ’e’, as the value of Re increases, the value of the Cddecreases. By analyzing the plot, for fixed value of the n=1, asthe value of ’e’ increases, the value of Cd increases respective ofthe values of Re. From the graph due to linear relation betweenpressure and Re, very less changes seen in Cd for different ’e’ athigh Reynolds numbers, but as decreasing value in Re, the changesin Cd is very high at low Re. so for a fixed value of n=1, the effectof ’e’ on Cd is more significant at low Re compared to high Re.

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Effect of ’e’ on Cdf and Cdp for fixed power-law indexFig. 3(b,c) shows the effect of eccentric factor on friction drag

coefficient, Cdf , as a function of the Reynolds number, Re, for fixedvalue of power law index, n=1. For fixed value of power law index,as the value of ’e’ increases, the value of Cdf decreases for all valuesof the Reynolds number. For fixed value of n=1, and ’e’ the fric-tion drag is linearly increases up to certain Reynolds number andsuddenly rising in the value of Cdf , then it reached a maximum Cdf

value at Re=10.so these sudden rise up in friction drag reduces therecirculation and flow separation at boundary of wall due to morefriction acting on the surface at low Re.

Fig. 3(d) shows a graph between Re and pressure drag Cdf/Cdp

for different ’e’ at constant power law index, n=1. For fixed valueof power law index, n=1, as the value of ’e’ increases, the valueof Cdf/Cdp increases for all values of the Reynolds number. Forfixed value of n=1, and ’e’, the friction drag is linearly increases upto certain Reynolds number and suddenly rise, then it reached amaximum Cdf/Cdp value at Re=10.

9.2 Drag Phenomena for shear-thinning power-law liquids fluid

Fig. 7 shows the effect of Re on the individual, total drag coeffi-cients and on the pressure to friction drag ratio of sphere particles(e = 1) for both Newtonian and power-law fluids. Regardless ofthe value of aspect ratio, both individual and total drag coefficientsdecreases as the value of Reynolds number increases. However,effects of rheological properties are more significant on pressure(form) drag coefficient rather on viscous and total drag coefficients.For all values of Reynolds number, as the value of power law index(n) increases, the form drag coefficient increases. Furthermore, forfixed values of aspect ratio (e=1), the total drag, pressure drag andratio between form drag and viscous drag increases as the value ofthe Reynolds number increases, but the effect is small for n=0.6and 0.8.

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10 CONCLUSION

The steady state behavior of Newtonian fluids past ellipsoidal parti-cles at different Reynolds numbers have been studied by employinga 2D, unsteady, segregated, single phase model in fluent. The for-mation of wake or a return- flow region at the rear of the bodydepends on Re and e at fixed power law index n=1. Irrespectivevalues of n and Re, recirculation and flow separation does not occurfor e≤0.75. For e≥1, as the value of Re increases, recirculation andflow separation is formed which grows in size and non-contiguous onthe surface of the body. The non-contiguous started from e=0.75,and becomes more complex with increasing the value of ’e’. Forfixed power law index and e, Cd is more at large values of Re than

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that at small values of Re. However as the value of e increased,Cd increases for all values of Re studied in this work. The localpressure drag and friction drag varies appreciably on the surface ofbody as the value of e increases for all values of Re. Similarly, themagnitude of wakes, recirculation and flow separation were studiedby analyzing the curves of streamlines and different drag coefficientsfor power law fluids. The effect of e on the Newtonian fluids of thelocal and overall kinematics was found to be more significant forsmall values of Re than that at large Reynolds numbers. The effectof rheology of power law fluids (n) on drag force for spherical parti-cles has been studied and found that there is a significant differencebetween Newtonian and power-law fluids.

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