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Wall Effects on Flow and Drag Phenomena of Spheroid Particles at Moderate Reynolds Numbers Nanda Kishore* and Sai Gu Energy Technology Research Group, School of Engineering Sciences, UniVersity of Southampton, Southampton, SO17 1BJ, United Kingdom
Two dimensional steady Newtonian flow past oblate and prolate spheroid particles confined in cylindrical tubes of different diameters has been numerically investigated. The flow and drag phenomena of confined spheroid particles are governed by the equations of continuity and conservation of momentum. These equations along with appropriate boundary conditions have been solved using commercial software based on computational fluid dynamics. Extensive new results were obtained on individual and total drag coefficients of spheroid particles, along with streamline contours, distributions of pressure coefficients, and vorticity magnitudes on the surface of spheroid particles as functions of the Reynolds number (Re), the aspect ratio (e), and the wall factor (λ) over the following range of conditions: 1 e Re e 200, 0.25 e e e 2.5, and 2 e λ e 30. For all values of the aspect ratio, as values of the Reynolds numbers and/or the wall factor increase, the length of recirculation wake increases. For fixed values of the aspect ratio and the Reynolds number, the increase in the value of the wall factor decrease both individual and the total drag coefficients. On the whole, regardless of the value of the aspect ratio, the wall effect was found to gradually diminish with the increasing Reynolds number and/or the wall factor. Finally, on the basis of the present numerical results a simple correlation has been proposed for the total drag coefficient of confined spheroid particles which can be used in new applications. 1. Introduction The flow of viscous Newtonian liquids past bluff bodies is an idealization of many industrial applications. Typical examples include flow in fixed and fluidized beds, slurry transportation of particulates in pipes, mineral slurries, waste streams, sewage sludge, coal-oil slurries transport, etc. In many of such applications, the stratification of solid-liquid suspensions is undesirable to ensure the trouble free transportation of the suspensions. In some food applications, during the thermal treatment of foodstuffs, the density-based stratification is detrimental to maintaining the final food quality. On the other hand, such separation is highly undesirable in order to maintain the homogeneity of the product such as fruit juices, soups, pharmaceutical products, liquid detergents, cleaning aids, etc. Therefore, regardless of the necessity of the stratification of suspensions, the knowledge of settling velocity (or drag force) is one of the important key parameter in design calculations of such solid-liquid contacting equipments. Admittedly, often arbitrary shaped particles and/or clusters of variety of solid particles are encountered in most engineering applications. However, the knowledge of the settling velocity of single regularly shaped particles such as spheres, spheroids, cylinders, cones, cubes, etc. provide the insight about the physics of the problem and would lead to the simplification of complex multiparticle systems. In addition, the effect of container wall on the drag force would further give the information about the interaction of the particle with the wall. Some basic applications of the particle settling in confined medium are falling ball viscometer, hydrodynamic chromatography, membrane transport, hydraulic transport of coarse solids in pipes, microfluidics, use of electric fields to enhance the rate of transport, etc.1-3 * To whom correspondence should be addressed. E-mail: nk2d09@ soton.ac.uk;
[email protected]. Tel.: +44-2380598301. Address: Energy Technology Research Group, School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, United Kingdom.
Because of the presence of the wall, the confining wall exert an extra retardation force on the particle, thus the sedimentation velocity of the particle is slowed-down. Therefore, the closer the particle-to-wall region is, the greater would be the retardation or wall effect; and consequently, all else being equal, the settling velocity of the particle is lower (hence larger is the drag force) in a confined fluid than in an unconfined medium. This information can be conveniently expressed in terms of the usual dimensionless parameters such as the Reynolds number, drag coefficient, the shape factor, and the wall factor. On the basis of a combination of experimental, theoretical, and numerical studies, reliable correlations/models for drag coefficients of unconfined and confined spheres and cylinders falling in Newtonian liquids are now available in the literature.1-8 On the other hand, for spheroid particles, although few experimental and numerical studies pertaining to the unconfined Newtonian fluid flow are available; there are no results reported on their wall effects even in the limit of zero Reynolds number. Thus, the aim of this work is to fill this gap in the literature. In this work, effects of the confining wall on the flow and drag phenomena of spheroid particles have been elucidated over wide ranges of pertinent dimensionless parameters. 2. Previous Work The pertinent literature on the flow and drag phenomena of unconfined and confined spheres, cylinders, cubes, etc. were already thoroughly reviewed elsewhere,1,4-8 hence, the literature pertaining to the unconfined and confined spheroid particles only are presented. It is believed that the first Stokes flow past unconfined spheroid particles was investigated by Oberbeck9 who proposed a correlation for the total drag coefficient. Later on, many semianalytical and theoretical studies were reported to obtain the flow and drag phenomena of spheroid particles in the small Reynolds number regime. Most of these studies either neglected or linearized the convective terms of the momentum
10.1021/ie1011189 2010 American Chemical Society Published on Web 09/02/2010
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equations and provided a correction shape factor for the total drag coefficient to account for the spheroid shape of the particle. Payne and Pell10 obtained the Stokes flow expression for a variety of axially symmetric bodies including oblate spheroids in unconfined Newtonian medium. For the case of rotating oblate and prolate spheroids, Breach11 generalized the results of Proudman and Pearson12 by solving the Navier-Stokes (N-S) equation of motion using the method of matched asymptotic expansions. In the range of finite Reynolds numbers (up to 4), Aoi13 obtained an analytical solution and proposed drag values of unconfined fixed oblate and prolate spheroid particles. The majority of semianalytical and numerical solutions of steady flow past unconfined, fixed, and rotating spheroids are based on series truncation methods, Fourier expansion methods, or Galerkin methods in conjunction with finite difference methods or finite element methods. By using the method of matched asymptotic expansions, Chwang and Wu14 have investigated the uniform traverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers. They reported the drag coefficient of prolate spheroid particles as functions of two Reynolds numbers, one is based on a semimajor axis and the other is based on a semiminor axis. Using a seriestruncation method, Alassar and Badr15 solved complete N-S equation and obtained the steady flow past oblate spheroids in the limit of small Reynolds number. Blaser16 calculated the forces acting on the surface of small rigid ellipsoidal particles immersed either in a constant, linear, simple shear, twodimensional straining or axisymmetric straining flow field in the creeping flow regime. Rimon and Lugt17 solved the complete N-S equation of motion by using a finite difference technique and obtained time-dependent viscous flow past an oblate spheroid of axis ratio 0.05 and 0.2 at Reynolds numbers of 10 and 100. Masliyah and Epstein18 and Comer and Kleinstreuer19 have numerically solved the steady flow past an oblate spheroid of axis ratio 0.2 and 0.5 for the range of Reynolds numbers up to 100 in order to elucidate the effects of the Reynolds number and the aspect ratio on streamlines and vorticity contours, surface pressure coefficients, and drag coefficients. Pitter et al.20 investigated the flow past thin oblate spheroids (of aspect ratio 0.05 and 0.2) by numerically solving the complete N-S equation at Reynolds numbers 0.1 and 100. Chang et al.21 have analytically and numerically obtained the axisymmetric viscous flow around ellipsoids of axis ratio 2:1 and a sphere in the range of Reynolds numbers of 100-3000 using the method of matched asymptotic expansion and a deterministic vortex method. Li et al.22 used the SIMPLE algorithm on a nonorthogonal boundary fitted staggered grid to investigate the interfacial transport characteristics of spheroid bubbles and solid particles up to Reynolds numbers of 100. Although their main emphasis was on comparisons of the drag, wake structures, and mass transfer rates between spheroid solid particles and spheroid inviscid bubbles, very little information on individual and total drag coefficients of spheroid particles has been provided. Tripathi et al.23 and Tripathi and Chhabra24 have numerically studied the power-law fluid flow past unconfined spheroid particles in the intermediate range of Reynolds numbers up to 100 and for aspect ratios from 0.2 to 5. Zimmerman25 has computed the singular behavior of drag coefficients when a disk blocks the cylindrical tube, by summing the perturbation series in radius ratio of the disk to the cylinder. Further Zimmerman25 computed the extension of regular series for the thin disk in broad side motion. Using the stabilization method on the basis of a variable direction scheme, Zamyshlyae and Shrager26 have numerically solved the complete N-S equations in the spherical coordinate
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system to obtain the steady viscous flow past slightly deformed spheres up to a Reynolds number of 100. Militzer et al.27 compiled the available literature on the experimental and numerical drag values of spheroid particles in the range of Reynolds number up to 200 and aspect ratios from 0.2 to 5 and proposed a correlation for the total drag coefficient of spheroid particles. However, for the case of a sphere, their results overestimate the drag values both at low and high Reynolds numbers. Recently, Wang et al.28 experimentally obtained the sedimentation characteristics of spheroid particles settling in static fluids in the range of Reynolds numbers up to 105. They observed that the Reynolds number has a logarithmic linear relationship with the Arrhenius number for spheroids settling with facets facing both upward and downward and their drag coefficients are larger than those of spheres. Chhabra et al.29 critically evaluated a selection of 19 independent experimental studies for estimating the drag coefficient of nonspherical but regular shaped particles settling in incompressible viscous fluids. Recently, Tran-Cong et al.30 experimentally obtained the steady state drag coefficients of various irregularly shaped particles and proposed a correlation for the total drag coefficient as a function of the Reynolds number, the particular circularity, and the ratio of the surfaceequivalent-sphere to the nominal diameters. On the other hand, there appears a large gap in the literature pertaining to the flow and drag phenomena of confined spheroid particles even in the creeping flow regime, let alone moderate to large Reynolds numbers. By using a finite element method, Sugihara-Seki31 investigated the motion of a rigid ellipsoidal particle freely suspended in a Poiseuille flow of Newtonian fluid through a narrow tube in the zero-Reynolds number limit. They found that a prolate spheroid will either tumble or oscillate in rotation, depending on the particle-tube size ratio, the axis ratio of the particle, and the initial conditions. Zimmerman32 treated different shaped particles including spheres and discs to obtain the additional drag on particles due to wall effects. Xu and Michaelides33 obtained the flow field and drag coefficient on one, two, and five tandem ellipsoidal particles inside a cylindrical tube up to a Reynolds number of 320 using a numerical scheme based on the Galerkin finite element technique and primitive variables formulation. Hsu et al.34 studied the wall effect on the total drag force acting on rigid spheroid particles settling in Carreau fluids in the range of low to moderate Reynolds number. Therefore, based on aforementioned discussion, it is safe to conclude that there are no studies reported in the literature on the flow and drag phenomena of confined spheroid particles in Newtonian fluids. Thus the aim of this work is to fill this gap in the literature in the following range of conditions: Reynolds number, Re, 1-200; aspect ratio, e, 0.25-2.5, and wall factor, λ, 2-30. 3. Problem Statement and Formulation Consider an oblate or a prolate located along the axis of a cylindrical tube at upstream distance Lu from the inlet and at downstream distance Ld from the outlet of the tube (Figure 1). Here the tube is assumed to be infinitely long so that the end effects are negligible. The spheroid particle is located at the center of the cylinder with Lu ) Ld ) L/2, where L () 100 m) is the length of the cylindrical tube. The tube wall surface is assumed to be smooth enough to avoid additional roughness effects. The diameter of the tube, Dt, is varied in order to account for different confinement effects. The wall effects are incorporated in the solver by varying the value of the wall factor (λ)
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Figure 1. Schematic representation of the flow past (a) a prolate and (b) an oblate spheroid in a cylindrical tube.
defined as the ratio between the diameter of the tube and the polar diameter of the particle, that is, λ ) Dt/(2b). The viscous fluid is steadily entering the tube at inlet with a velocity V (m/ s) and pressure P0 (Pa) and leaving the tube at outlet with pressure P (Pa). In other words, this study is equivalent to the investigation of the settling behavior of oblate and prolate particles in cylindrical tubes of different diameters filled with a Newtonian liquid. The larger the tube diameter is, the closer is the settling velocity (or drag force) of the particle to the case of unbounded particle. As the diameter of the tube decreases, the settling of the particle is influenced by cylinder walls and thus the velocity decreases and drag experienced by particle increases as compared to the case of unbounded particle. Thus here in this study our aim is to investigate the wall proximity on drag phenomena of spheroid particles in cylindrical tubes, as has been investigated by other researchers for the case of spheres2,3 and ellipsoidal particles,33 rather than what is happening at the wall. The governing equations for this flow problem are the continuity and the Navier-Stokes equations as written below: Continuity equation
∇·V ) 0
Navier-Stokes equation DV ) -∇P + µ∇2V F Dt
At the surface of the particle, the standard no-slip condition has been used: Vx ) 0, Vy ) 0. At the outlet, the pressure outlet boundary condition has been used so that the conditions upstream of the exit shall not be influenced: ∂P/∂x ) 0; ∂Vx/∂x ) 0; ∂Vy/∂x ) 0. Along the tube wall, in order to account for the effect of the wall, a stationary wall with no-slip boundary condition has been imposed. Along the axis of tube, the symmetry boundary condition has been used: ∂Vx/∂y ) 0; Vy ) 0. Equations 1 and 2 along with the above-mentioned boundary conditions provide the complete theoretical frame to map the entire flow domain in terms of Vx, Vy, and P for the present range of conditions. These equations have been numerically solved using Fluent as described in the next section. Once the fully converged steady-state velocity and pressure fields are obtained, these are further processed to evaluate streamline patterns, vorticity contours, individual and total drag coefficients, distributions of the pressure coefficient, and the vorticity magnitude on surface of spheroid particles, etc. Here the total drag coefficient, Cd, is defined as
(1) Cd )
2Fd FV2Ap
) Cdp + Cdf
(3)
(2)
where V is the velocity vector (m/s), F is the density of the fluid (kg/m3), µ is the viscosity of the fluid (kg/m · s) and P is the pressure (Pa). The appropriate boundary conditions for this flow problem can be written as follows: At the inlet, the uniform axial velocity is prescribed, that is, Vx ) V, Vy ) 0.
where Fd is the drag force acting on the particle, Ap is the area of the particle, and Cdp and Cdf are the pressure and frictional components of the total drag coefficient. In this work, the derived quantities have been reported as functions of the Reynolds number (Re), the aspect ratio (e), and the wall factor (λ). The Reynolds number is defined based on the polar radius of the spheroid as 2bVF/µ and the aspect ratio is the ratio between the polar and equatorial radii of the spheroid, e ) b/a.
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4. Numerical Solution Methodology Because of the complicated nature of the flow, theoretical analyses are typically limited to the cases of Reynolds numbers approaching either zero or infinity. Further, in recent years, experimental techniques have become sophisticated; on the other hand, due to the advent of faster computers, reasearchers now have a great interest in numerical techniques used to obtain the solutions of such complicated flow problems. Thus, in recent years, many numerical methodologies have been developed along with mixed CFD/analytical methodologies.35 Also commercial softwares have been developed like Fluent, CFX, Polyflow, Star-CD, etc. which can provide reliable solutions with faster convergence. Therefore, it has been decided to numerically obtain the solution of this flow problem by using commercial software based on Computational Fluid Dynamics (CFD), Fluent 12.0, in conjunction with a mesh generating software Gambit. In the Gambit, first various boundaries like inlet, outlet, tube wall, central axis, and the shape of the particle have been generated with specified dimensions. Using these edges, the computational flow domain has been generated and appropriate boundary conditions have been prescribed to respective boundaries. To capture the boundary layer thickness with adequate accuracy, the number of interval counts on the axis edges were chosen with a specified ratio so that a larger number of grid points cluster near the surface of the particle. Finally, the triangular-type structured mesh has been generated in the entire domain using the node points of various edges. Thus generated mesh has been exported to the Fluent. The fluid properties read from the Fluent database, and quantitative boundary conditions have been specified in order to meet the required values of the Reynolds number. The SIMPLE solution method with first order upwind scheme for convective terms has been chosen. The tolerance residual values for the convergence of the continuity, x-velocity, and y-velocity components of momentum equations have been set to 10-10 in order to make sure that the solution is completely converged. Finally the solution procedure initialized from the inlet and continued calculations until the steady-state velocity and pressure fields were obtained. Once the fully converged velocity and pressure fields were obtained, these values were further used to evaluate integral quantities such as the individual and total drag coefficients, distributions of surface pressure coefficients, and surface vorticity magnitudes, etc. 4.1. Numerical Details. The accuracy and reliability of such numerical solutions are prudent upon the appropriate choice of the numerical domain, optimum grid, and the solution procedure. Since the spheroid particles are confined in cylindrical tubes of various diameters, the size of the computational domain is dependent on the aspect ratio of the particle and the diameter of the tube of interest. Let us say, for a prolate of aspect ratio e ) 1.5 and for wall factors of λ ) 5 and λ ) 10 the diameters of respective cylindrical tubes are Dt/2 ) 3.75 m and Dt/2 ) 7.5 m. Therefore, grid refinement tests should be carried out for various combinations of the Reynolds number, aspect ratio, and wall factor. In general, as the value of the Reynolds number increases, the boundary layer becomes thin on the surface of the particle. Thus the grid refinement study was carried out at large value of the Reynolds number of this study, that is, at Re ) 200 for extreme values of aspect ratio (e ) 0.25 and e ) 2.5) and wall factors of λ ) 3 and λ ) 30. For each case, three grids have been used with the following nodes on the surface of the particle, upstream axis, and downstream axis, respectively; (Grid 1) 200, 200, 200; (Grid 2) 300, 300, 300; (Grid 3) 350, 350, 350. From Table 1, it can be seen that the relative
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Table 1. Effect of Grid Size on Cd at Re ) 200 for Extreme Values of e and λ wall factor: λ ) 3
wall factor: λ ) 30
Grid 1 Grid 2 Grid 3 Grid 1 Grid 2 Grid 3 oblate spheroid: e ) 0.25 prolate spheroid: e ) 2.5
0.811 5.948
0.775 5.762
0.774 5.674
0.363 2.481
0.348 2.425
0.349 2.438
Table 2. Comparison of the Present Cd Values of Unconfined Spherical Particles (e ) 1) Re
Le Clair et al.36
Saboni et al.37
Kishore et al.38
present
200 100 50 20 10 1
0.772 1.096
0.776 1.084 1.589 2.768 4.424 27.55
0.779 1.094 1.584 2.739 4.344 27.708
0.772 1.089 1.577 2.720 4.310 27.467
2.736 4.337 27.37
Table 3. Comparison of the Present Cd Values of Unconfined Spheroid Particles at Re ) 0.01
present Happel and Brenner5 Tripathi et al.23
e ) 0.2
e ) 0.5
e)1
e)2
e)5
2188 2068 2126
2259 2173 2231
2456 2400 2457
2956 2889 2951
4309 4284 4382
Table 4. Comparison of the Present Cd Values of Confined Spherical Particles (e ) 1) λ)3
λ)5
Re
Wham et al.
present
Wham et al.
1 10 20 50 100
126.934 14.235 8.252 4.574 3.149
120.681 13.648 7.959 4.374 2.959
81.854 11.412 7.266 4.349 3.037
3
3
λ ) 6.4 present Wham et al.3 present 79.775 11.031 6.938 3.965 2.647
72.253 10.575 6.906 4.230 2.967
71.521 10.719 6.782 3.860 2.553
differences between the drag coefficients obtained by Grid 2 and Grid 3 are less than (0.6% except for the case of e ) 2.5 and λ ) 3. Thus Grid 2 with 300 nodes on the surface of the particle, 300 nodes on both Lu and Ld, and 100 nodes on both inlet and outlet edges was used for computations of all combinations of the Reynolds number, aspect ratio, and wall factor. 5. Results and Discussion The nondimensionalization of the governing equations and the boundary conditions suggest the individual and total drag coefficients are functions of the Reynolds number, the aspect ratio, and the wall factor. In this study, this functional relationship is elucidated over wide ranges of parameters: the Reynolds number, Re ) 1, 10, 20, 50, 100, 200; the aspect ratio, e ) 0.25, 0.5, 1, 1.5, 2, 2.5; and the wall factor, λ ) 2, 3, 5, 10, 20, 30. 5.1. Validation. Before presenting new results, it is customary to establish the reliability and accuracy of results by benchmarking present drag coefficients with literature values for various possible cases. Table 2 shows the comparison of the present values of total drag coefficient of unconfined solid spheres (i.e., e ) 1) with literature values36-38 at different Reynolds numbers ranging from 1 to 200. The present results for unbounded solid spheres are found to be within 2-3% of literature values. Table 3 shows the comparison between the present drag values and analytical and numerical results by Happel and Brenner5 and Tripathi et al.23 respectively, at Re ) 0.01 (in this case the Reynolds number is defined based on the equivalent diameter of the spheroid). Here too excellent agreement is found between the two results. Table 4 shows the comparison pertaining to wall effects on solid spherical particles, and it is found that the present results are a good match with
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Figure 2. Streamline contours of spheroid particles at Re ) 20 (upper half, solid lines) and Re ) 200 (lower half, dashed lines) for different values of the wall factor: (a-c) e ) 2.5, (d-f) e ) 1.5 and (g-i) e ) 0.5.
those of Wham et al.3 for Reynolds numbers up to 100 and the wall factor λ ) 3, 5, 6.4. However, as the values of the Reynolds number and/or the wall factor increases, the difference between two results increases. In general, the differences between numerical values obtained using different numerical methodologies can be attributed to the method of solution, domain, and grid used. On the other hand, the difference between the experimental and numerical results may be due to the wall roughness and orientation/rotation of the particle, etc. For instance, in microfluidic applications, the major uncertainty due to the microfabrication nonidealities is actually the wall roughness/position, which leads to 3-5% discrepancies between the measured velocity fields by micro-PIV and CFD predictions.39 Thus the differences of this order of magnitude are not at all uncommon in such numerical studies and are frequently ascribed to the numerical procedure, grid adopted, etc.38,40 Therefore, on the basis of extensive comparisons shown in Tables 2-4, it is believed that the present results are accurate and reliable within (3-5%. 5.2. Streamline Patterns. Figure 2 represents the combined effects of the Reynolds number, aspect ratio, and wall factors on streamline contours of spheroid particles. The upper-half of the particle depicts the streamlines at Re ) 20 (solid lines) while the lower-half depicts at Re ) 200 (dashed lines). In general for an unconfined particle the velocity field decays progressively and ultimately the velocity gradients vanish far from the particle. In contrast to unbounded particles, for confined particles, due to the imposition of the no-slip boundary condition both at the surface of the particle and at the wall, the velocity gradient will not vanish monotonically, thus the fluid experiences somewhat higher levels of shearing in the confined region. Therefore, in the case of confined particles, the length of the recirculation wake is smaller than in the case of unconfined particles.
However, for both unconfined and confined cases, as the value of the Reynolds number increases, the lengths of the recirculation wake increases. From Figure 2a-c it can be seen that for a prolate spheroid of aspect ratio, e ) 2.5, as the value of the wall factor increases (i.e., the decreasing retardation effect), the length of the recirculation wake of the flow increases due to the reduced dissipation because of increased distance between the particle and the wall. This observation can be made for both values of the Reynolds numbers, that is, for Re ) 20 (upperhalf) and Re ) 200 (lower-half). Regardless of the value of the wall factor, as the value of the Reynolds number increases, the length of the recirculation wake increases. The same observations can be made for other values of the aspect ratio, that is, for e ) 1.5 (Figure 2d-f) and for e ) 0.5 (Figure 2g-i). However, there is no flow separation for spheres and oblate spheroids for Re e 20. On the other hand, regardless of the value of the wall factor, for a fixed value of the Reynolds number, as the value of the aspect ratio decreases, the length of the recirculation wake decreases. Therefore, in summary, regardless of the value of the wall factor, there is no flow separation for spheres (e ) 1) and oblate spheroids (e < 1) for Re e 20; however, for other combinations of Re, λ, and e, as the value of the Reynolds number and/or the aspect ratio and/or the wall factor increases, the length of the recirculation wake increases. 5.3. Distribution of Surface Pressure Coefficients. Figure 3 shows the combined effects of the aspect ratio and the wall factor on the distribution of the pressure coefficients, defined as 2(Po - P∞)/(FV2), on the surface of the spheroid particles at Re ) 200 (a-c) and Re ) 20 (d-f). For a prolate of aspect ratio e ) 2.5 for all values of the wall factor, the pressure coefficient decreases from its initial value at the front stagnation point to a certain lower value almost on the top of the particle
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Figure 3. Effects of the aspect ratio and the wall factor on the distribution of the pressure coefficient on the surface of the particle at (a-c) Re ) 200 and (d-f) Re ) 20.
surface, then starts to increase up to the point of flow separation (due to the recovery of pressure), and then again starts rising further to improve the recovery because of the recirculation wake at the rear end of the surface of the particle. The recovery is poor for a small value of wall factor, for instance, λ ) 2; however, the recovery improves as the value of the wall factor gradually increases. For a fixed value of the wall factor, as the value of the aspect ratio decreases gradually from e ) 2.5 to e ) 0.5, the pressure recovery improves. Thus, the pressure drag coefficient is large for large values of the aspect ratio and/or small values of the Reynolds number and/or small values of the wall factor. Similar observations can be made from other values of the aspect ratio and the Reynolds number. In summary, regardless of the values of the aspect ratio and the wall factor, the pressure recovery improves as the value of the Reynolds number increases. For fixed values of the Reynolds number, the pressure recovery is good for spheroids of small aspect ratio and/or large values of the wall factor. 5.4. Distribution of Surface Vorticity Magnitudes. Figure 4 shows the combined effects of the aspect ratio and the wall factor on the distribution of the vorticity magnitude (|∇ × V|) on the surface of the spheroid particle at Re ) 200 (a-c) and Re ) 20 (d-f). For a prolate of aspect ratio, e ) 2.5, for all values of the wall factor, the vorticity increases from its initial value at the front stagnation point to a certain finite value up to almost the top of the particle, then it starts to decrease to a minimum value up to the point of separation, and then it gains momentum to certain a finite value before it relaxes at the rear stagnation point. The obvious reason for this behavior is that the shear rate gradually increases from the front stagnation point to a certain maximum value (corresponding to the position of peak of the primary hump) and then gradually decreases up to the point of separation. The peculiar behavior of forming a secondary hump in the rear half of the particle is because of
the flow separation or due to the formation of recirculation wake as seen in Figure 2. Further, for fixed values of the aspect ratio and the wall factor, as the Reynolds number increases, the shear rate increases; thus, the vorticity magnitude is large for large values of the Reynolds number. As the value of the wall factor increases, the shear rate decreases; thus, the heights of primary and secondary peaks are decreased due to the reduced dissipation (because of the increased flow distance between the particle and the wall). Thus, the friction drag coefficient is large for small values of the wall factor and/or for large values of the Reynolds number. For all other values of the Reynolds number and the wall factor, similar observations can be made for other prolate spheroid particles or spherical particles. However, for oblate particles the value of the vorticity shoots up right at the front stagnation point and then follows the similar behavior that is true for prolate spheroid particles. Thus, in summary, for all values of the Reynolds number and aspect ratio, e g 1, the primary and secondary peak heights of vorticity distribution decreases as the value of the wall factor increases and causes the friction drag coefficient to decrease. However, for oblate particles the life of the primary peak is very short and then the vorticity drastically falls to some finite minimum value at the point of separation before it starts rising due to the formation of recirculation wake in the rear half of the particle. 5.5. Drag Phenomena. In general, the drag force on a confined particle is higher than on the unconfined particle due to additional dissipation because of wall retardation effects. Figure 5 shows the combined effects of the aspect ratio, Reynolds number, and wall factor on the drag experienced by the spheroid particles. The characteristic nature of Cd vs Re is found in the case of oblate and prolate spheroid particles for all values of the wall factor. Thus, regardless of the value of the aspect ratio and the wall factor, the drag coefficient decreases
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Figure 4. Effects of the aspect ratio and the wall factor on the distribution of the vorticity magnitude on the surface of the particle at (a-c) Re ) 200 and (d-f) Re ) 20.
Figure 5. Effects of the aspect ratio, the wall factor, and the Reynolds number on the total drag coefficient of spheroid particles.
as the value of the Reynolds number increases. For fixed values of the wall factor and the Reynolds number, the drag coefficient is large for prolate particles as compared to spheres and the opposite trend has been observed for oblate spheroid particles. The obvious reason for this behavior is that the front area of
prolate particles is large, thus, the resistance to the flow is more; hence prolate particles experience more drag as compared to spheres and oblate spheroids. For all values of the aspect ratio and the Reynolds number, the value of the drag coefficient decreases as the value of the wall factor increases. This is
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Figure 6. Effects of the aspect ratio, the wall factor, and the Reynolds number on the ratio between the pressure and the friction drag coefficients of spheroid particles.
because of the reduced dissipation due to increased flow region between the particle and the wall. Thus, the particle settles with ease as the wall moves away from the particle. Figure 6 shows the combined effects of the aspect ratio, Reynolds number, and wall factors on the ratio between the pressure and friction drag coefficients of spheroid particles. Regardless of the value of the aspect ratio and the wall factor, the drag ratio increases as the value of the Reynolds number increases. This is because the pressure and frictional contributions in the total drag coefficient for an unconfined particle in the creeping flow region are 1/3 and 2/3, respectively; thus, the drag ratio is 1/2. The drag ration increases as the value of the Reynolds number increases because the decrement of viscous forces relative to the increment of convective forces is rational at least in the axisymmetric flow regime. The same reason holds true even for confined particle, though the ratio is quantitatively large for closely confined particle and it decreases as the confinement reduces. Furthermore, for fixed values of the Reynolds number and the wall factor, the value of the drag ratio is large for prolate spheroid particles as compared to spherical particles and the opposite trend has been observed for oblate spheroid particles. This behavior can again be attributed to the large frontal area of prolate spheroids and the competition between the decrement of viscous forces and the increment of convective forces in the axisymmetric flow regime. On the other hand, for oblate spheroids of aspect ratio, e < 0.5, for fixed value of the wall factor, the value of the drag ratio is almost insensible to values of the Reynolds number as the shape of highly oblate spheroids offers very little resistance to the flow. However, effect of the wall factor on the drag ratio of oblate spheroid particles is significant. In summary, for fixed values of the aspect ratio and the Reynolds number, the decrease in the value of the wall factor promotes both the total drag coefficient and the ratio between the pressure and friction drag coefficients.
Finally, it is useful to develop a model for the total drag coefficient which can be used to predict the rate of settling of spheroid particles in confined enclosures in new applications. Thus, based on the present numerical results, the following drag model has been proposed: Cd )
12 λ + 0.79 Re λ + 1.81
(
-4.16
)
[1 + 0.304Re0.685e1.1 + 0.15Re-0.5e3.3 + 2.1e0.4] (4)
The drag coefficients obtained by above equation are in excellent agreement with the present numerical results (216 data points). The above correlation reproduces the present numerical results with an average relative error of (9.2% which rises to a maximum of (37.5% in the range of 1 e Re e 200, 0.25 e e e 2.5, and 3 e λ e 30. Figure 7 shows the comparison between the present numerical drag coefficients and those obtained by the present drag model (eq 4) and there are no discernible trends observed for any combinations of Re, λ, and e. Thus, this drag model can be used to estimate the drag coefficient (or settling velocity) of confined spheroid particles in cylindrical tubes filled with Newtonian fluids in new applications. 6. Conclusions The effect of the wall confinement on the flow and drag phenomena of spheroid particles in cylindrical tubes has been numerically investigated using a commercial CFD solver, Fluent. Regardless of the value of the wall factor, there is no flow separation for spheres (e ) 1) and oblate spheroids (e < 1) for Re e 20; however, for other combinations of Re, λ, and e, as the value of the Reynolds number and/or the aspect ratio and/ or the wall factor increases, the length of the recirculation wake increases. For fixed values of the aspect ratio and the Reynolds
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Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010 Vx ) x-component of the velocity, m/s Vy ) y-component of the velocity, m/s Greek Symbols λ ) wall factor (Dt/(2b)), dimensionless µ ) viscosity of the fluid, kg/(m · s) F ) density of the fluid, kg/m3
Literature Cited
Figure 7. Comparison between the present numerical and correlated drag coefficients.
number, the pressure recovery improves as the value of the wall factor increases. For all values of the Reynolds number and aspect ratio, e g 1, the primary and secondary peak heights of vorticity distribution decrease as the value of the wall factor increases and causes the friction drag coefficient to decrease. However, for oblate particles the life of the primary peak is very short and then the vorticity drastically falls to some finite minimum value at the point of separation. For fixed values of the aspect ratio and the Reynolds number, the decrease in the value of the wall factor promotes the individual and the total drag coefficients. On the whole, regardless of the value of the aspect ratio, the wall effect was found to gradually diminish with the increasing Reynolds number and/or the wall factor. Finally, on the basis of the present numerical results a simple correlation has been proposed for the total drag coefficient of confined spheroid particles which can be used in new applications. Acknowledgment The authors gratefully acknowledge the financial support from Royal Commission for the Exhibition of 1851 Research Fellowship Award, EU FP7 SIMUSPRAY Project (Grant No. 230715), UK EPSRC Grant EP/G034281/1, and LeverhulmeRoyal Society Africa Award. Nomenclature a ) equatorial radius of the spheroid particle, m Ap ) area of the particle, m2 b ) polar radius of the spheroid particle, m Cd ) total drag coefficient, dimensionless Cdf ) friction drag coefficient, dimensionless Cdp ) pressure drag coefficient, dimensionless Dt ) diameter of the tube, m e ) aspect ratio () b/a), dimensionless Fd ) drag force, N L ) length of the tube, m Ld ) downstream distance, m Lu ) upstream distance, m P ) pressure, Pa Re ) Reynolds number, dimensionless V ) velocity vector, m/s
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ReceiVed for reView May 18, 2010 ReVised manuscript receiVed August 5, 2010 Accepted August 19, 2010 IE1011189
