ARTICLE pubs.acs.org/IECR
Effects of Particle Diameter and Position on Hydrodynamics around a Confined Sphere Suresh Krishnan and Kannan Aravamudan* Department of Chemical Engineering, Indian Institute of TechnologyMadras, Chennai 600036, India ABSTRACT: The effect of confinement of a sphere within a tube on the hydrodynamics of Newtonian fluid flow around the solid surface is investigated. The ratio of particle diameter to tube diameter (blockage ratio), the ratio of sphere distance from the tube axis to the tube diameter (eccentricity), and the fluid flow rate were the parameters of this study. Computational fluid dynamics (CFD) simulations were carried out to obtain the flow field around the sphere from which the angle of boundary layer separations, as well as drag and lift coefficients, were obtained. The pressure distributions and friction factor along the sphere surface are also reported. To estimate the effect of confinement and eccentricity in position of the particle, with respect to the tube axis, the results are compared with the classical case involving the flow of an unbounded fluid over the sphere. The drag coefficient diminished when the particle was positioned eccentrically with respect to the tube axis at very low particle Reynolds number (Rep). However, the drag coefficient increased with increasing eccentric positions at higher Rep. At eccentric particle positions, boundary layer separation occurred earlier from the upper hemisphere while, depending on the eccentricity, it was delayed or nonexistent in the lower hemisphere. The difference in pressure between the lower and upper hemisphere regions led to incursion of fluid into the upper hemispherical region. Correlations are proposed for the drag and lift coefficients in the range of 0.1e Rep e 500 for different possible eccentricities and blockage ratios. The results from this work will be of relevance in applications such as fluidization and aseptic food processing.
1. INTRODUCTION Movement of a fluid past a sphere has been widely investigated, because of its importance in both engineering applications and basic research. The hydrodynamics are complicated by various factors such as particle size, fluid velocity, rheological properties of the fluid, and the velocity profile of the fluid entering the tube. While the sphere suspended in an uniform flow field has been widely reported in the past,17 relatively few studies (e.g., Wham et al.8 and Shahcheraghi and Dwyer9) have investigated the hydrodynamics around a sphere immersed in a confined enclosure wherein the fluid approaching the particle has a parabolic velocity profile, such as that in laminar flow. Furthermore, the effects of blockage ratio (the ratio of the particle diameter to the tube diameter) and particle eccentricity (offset of the sphere’s center from the tube axis, expressed as a fraction of the tube diameter) may also begin to influence the hydrodynamics and have been only scarcely investigated. In the design of processes involving momentum transfer between the fluid and particles confined in tubes, important parameters such as drag and lift coefficients that take into account the effects of particle position and diameter were unavailable over a wide range of conditions for accurate modeling work. This may constrain accurate design of industrially important processes. One important example is in the field of aseptic food processing, where the relatively slow-to-heat food particles are sterilized by heating the carrier fluid inside a holding tube.1013 In these applications, the kinetic parameter of microbial destruction is highly sensitive to temperature. Variations in heating duration and heating rate of the particle may significantly affect the outlet product sterility. Particles with very small residence times inside the holding tubes may be sterilized inadequately, while particles with high residence times may undergo excessive heating and face a loss of nutritive r 2011 American Chemical Society
value. The temperaturetime history of the food particle depends upon the particle residence time, the fluid-to-particle heattransfer coefficient, and the thermal diffusivity of the particle. In the design of aseptic food processing system, the residence time and heating rate of the particle should be known accurately for optimized design. This work addresses the effect the blockage and eccentricity on hydrodynamic parameters such as drag and lift coefficients, which influence the residence time distribution of the particle in the holding tube. The effect of blockage on heat transfer has been discussed by Suresh and Kannan.14,15 They observed a strong effect of confinement on the drag coefficients, especially at low particle Reynolds number (Rep) values. Alhamdan and Sastry16 attributed higher lift forces that were created by moreviscous fluids to shorter residence times of particles inside the holding tube. In water, higher lift forces on particles were observed. Alhamdan and Sastry16 also observed particles at eccentric positions, with respect to the axis of the holding tube. In aseptic food processing applications, the particle size may become comparable with the holding tube size, the latter being typically on the order of 5.08 cm. The particle sizes themselves may vary from 1.28 cm to 2.23 cm. Consequently, the blockage ratios range between 0.25 and 0.44. The Rep values range between 1 and 200 and the corresponding fluid Reynolds numbers (Ref) range between 0.0058 and 798.17 Examples of solid particles in aseptic food processing include potatoes, carrots, turkey, green Special Issue: Ananth Issue Received: January 14, 2011 Accepted: June 2, 2011 Revised: May 26, 2011 Published: June 02, 2011 13137
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Industrial & Engineering Chemistry Research peas, and parsnip particles suspended in various Newtonian and non-Newtonian fluids (water, Carboxy Methyl Cellulose (CMC), and pectin solution).16,18 The effect of blockage and eccentricity on the drag and lift forces acting on the particle are investigated in this study. Accurate expressions for drag and lift coefficients may better describe the forces acting on the particle in computational fluid dynamics (CFD)-based simulations. Model studies based on a single-sphere configuration are also relevant to the development of appropriate frameworks for the analysis of multiparticle systems as encountered in real-life applications.19 Single food particle residence time distribution studies inside holding tubes have been carried out.2022 Other applications of relevance include fluidization. Wham et al.8 have studied the effect of blockage ratio in the case of fluidized spheres. The blockage ratios studied by Wham et al.8 range between 0.08 and 0.7 for Rep values up to 100. It is not possible to easily ascertain the extent of inaccuracy in using standard drag correlations that are applicable for bodies immersed in an unbounded fluid domain where the fluid approaching the sphere has a flat velocity profile, to actual situations involving a bounded fluid domain and the fluid enters it with a parabolic velocity profile. The scope of this work is to understand the underlying hydrodynamics behind the influence of particle blockage and eccentricity in position on flow patterns, pressure distribution, and boundary layer separation. Correlations are developed for drag and lift coefficients in confined flows.
2. BACKGROUND AND SCOPE The hydrodynamic effects of the flow field around the sphere are quantified in terms of the drag and lift coefficients, which, in turn, are dependent on the distribution of the pressure and viscous forces around the spherical particle.23 Furthermore, the drag coefficient also plays a more direct role in certain applications such as aseptic food processing, where the residence time of the food particle that is being heated in a tube is dependent on the forces acting on the particle. Early studies on the effect of the tube wall on the particle drag were mainly confined to creeping flows.2426 The characteristics of creeping flow have been outlined in McCabe et al.27 Bohlin’s28 correction factor for the drag force applied under conditions where the particle moved along the centerline of the tube with a velocity that was different from the approach velocity of the fluid. The fluid was in laminar flow. The correction factor was applicable for very small blockage ratios and very low Reynolds numbers (based on the sphere velocity). Happel and Brenner29 considered creeping flow of a sphere along the axis of a cylindrical tube filled with a fluid. Their drag force prediction that also accounted for the eccentricity of the particle position, with respect to the tube axis, was applicable only for very small blockage ratios. Fayon and Happel30 provided a corrected correlation for the drag coefficient at low particle Reynolds number range (Rep = 0.140) when the sphere was suspended inside a tube. Blockage ratios in the range of 0.1250.3125 and different eccentricities in particle position (0.125, 0.25) were considered. Their correlation assumed that the effect of the tube wall was additive with that due to inertia. Their correlation was found to be increasingly inaccurate at higher Reynolds numbers. Wakiya,25 as well as Haberman and Syre,26 provided relations by incorporating higher-order terms to predict the drag force on a sphere in the creeping flow limit. A general trend of increase in drag coefficient with increasing blockage ratio was observed for the particle positioned at the center of the
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tube. Johansson31 numerically studied the hydrodynamics around a sphere positioned at the axis of a cylindrical tube with the fluid in Poiseuille flow. For the blockage ratio of 0.1 that was studied, the results indicated that the wall effects were negligible for the tubediameter-based Reynolds number in excess of 50. Tozeren32 considered steady viscous flow at low Reynolds numbers past an eccentrically positioned sphere in a cylindrical tube. The eccentricities, with respect to the tube axis, were much smaller than unity and blockage ratios up to 0.7 were studied. The drag force was expressed in terms of coefficients that were functions of both eccentricity and blockage. Tozeren32 observed that the drag diminished as the sphere moved away from the axis, and this reduction was more significant as the blockage ratio increased. Ambari et al.33 experimentally obtained the drag acting on a sphere moving relative to a fluid in a cylindrical tube. Both blockage and eccentricity of the sphere relative to the tube axis were considered at very low Reynolds number (∼104). The drag was observed to decrease and pass through a minimum when the sphere moved away from the tube axis. However, lift coefficients were not reported in these studies. Oh and Lee4 studied, experimentally and numerically, the Newtonian fluid flow past a sphere in a tube. Flat velocity profile was specified at the inlet and exit boundaries. A restricted range of Reynolds numbers (20130) and blockage ratios (0, 0.5, and 0.74) were considered. Increased tube blockages were found to lead to higher drag coefficients. Wham et al.8 numerically studied the wall effects on drag force on a sphere positioned in a cylindrical tube. The Rep value in their work was extended to 100, and the effect of varying the blockage ratio from 0.08 to 0.7 was studied. The drag force results computed by Wham et al.8 agreed with those of Wakiya,25 Haberman and Sayre,26 and Fayon and Happel30 for blockage ratios of e0.3. Wakiya’s25 solution tended to underestimate the drag at blockage ratios of >0.4. The Haberman and Sayre26 solution followed the results of Wham et al.8 until a blockage ratio of 0.5 are attained. Wham et al.8 showed that, for Rep values up to 100, drag forces in narrow tubes (implying higher blockage ratios), although higher when compared to lower blockage ratios, were found to increase only slowly with Reynolds numbers. At low to moderate blockages, the drag force changed more rapidly for low to moderate values of the Reynolds number (for instance, 0.04 began to influence the pressure drag and delayed the boundary layer separation around the sphere. Suresh and Kannan14 investigated the effect of blockage ratio on both hydrodynamics and heat transfer around a confined sphere positioned centrally in a tube. A drag coefficient correlation was developed in terms of blockage and Rep values up to 500. They concluded that the form drag was affected more than the viscous drag by the effects of blockage. Higher blockage ratios were observed to delay the boundary layer separation. In the literature, the work has been carried out mainly over a restricted range of Reynolds numbers and a centrally positioned sphere.4,8 The conditions in these studies are summarized in Table 1. In the limited body of work that have considered eccentricity,9,32,33 it has been observed that, when the particle is 13138
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Table 1. Summary of Literature for a Single Sphere in Tube Flow Shahcheraghi and
Fayon and
Dwyer9
Happel30
Wham et al.8
Suresh and Tozeren32
Ambari et al.33
Kannan14
blockage ratio, BR
0.080.7
0.2, 0.4
0.1250.3125
0.0010.7
0.120.6
0.020.5
eccentricity, E
centrally positioned
0.2
0.125 and 0.25
∼0
0 to ∼1
0
Rep e 100
25 and 125
0.120
Stokes law regime
Stokes law regime
Rep e 500
transient/steady
steady
steady
steady
steady
steady
transient
type of solution
2D simulations
3D simulations
experimental
perturbation solution
experimental
3D simulations
inlet velocity
parabolic velocity
parabolic velocity
parabolic velocity
parabolic velocity
parabolic velocity
parabolic velocity
sphere Reynolds number range, Rep)
boundary condition boundary layer
profile yes
profile no
profile
profile
no
profile
no
profile
no
yes
separation considered?
displaced only slightly from the axis of the tube, the drag coefficient decreases at a given blockage ratio. However, when the particle is positioned very close to the wall of the tube, the drag coefficient started to increase.33 These observations have been made at very low Reynolds numbers. Only Shahcheraghi and Dwyer9 reported the effect on lift for an eccentrically positioned sphere. Lareo et al.34 conducted a series of experiments to study the behavior of single particles immersed in either water or CMC solutions. They observed maximum relative velocities between the carrier fluid and particles to be on the order of 102 m/s. The corresponding Rep value is calculated to be ∼500. Therefore, it becomes essential to predict the drag and lift coefficients over a wider range of Rep values (i.e., up to 500 at different blockage ratios and eccentricities). It becomes necessary to explain the effects of blockage and eccentricity at higher Reynolds numbers. In the Lagrangian particle tracking, where the particles are tracked for their position, velocity, and residence time, accurate expressions for drag and lift forces acting on the particles are required that take into account the particle’s diameter, relative to the tube, as well as it is position relative to the tube axis. Unluturk and Arastoopour13 estimated, through simulations, the residence times of spherical particles suspended in a 0.3% CMC solution flowing through a tube. They used drag coefficient correlations that were only functions of the Reynolds numbers. These correlations did not seem to consider either the significant blockage ratio (0.21) of the particle injected into the tube or its significant radial deviation from the tube’s axis. In the current study, CFD simulations are used to study the effects of both blockage and eccentricity in position when there is laminar flow past a sphere fixed inside a tube and the fluid approaches the particle in well-developed Poiseuille flow. A broader range of tube and particle Reynolds numbers in the laminar regime is considered in this work. The hydrodynamic results are compared with previous results that considered blockage for the purpose of validation. Furthermore, the results obtained here are also compared with the classical results involving a fluid approaching an unconfined sphere with a uniform velocity (i.e., with a flat velocity profile). This was done in order to illustrate the errors involved in using standard correlations developed for classical solutions to cases involving particles with high blockage ratios, fluid approaching the particle with a nonuniform (i.e., parabolic) velocity profile and different positions of the sphere, with respect to the tube’s axis.
Table 2. Physical Properties of Water at 303 K SI. number 1
property
parameter
value
density
F
997 kg/m3
2
specific heat capacity
Cp
4174 J/(kg K)
3
thermal conductivity
k
0.623 W/(m K)
4
viscosity
μ
7.64 104 Pa s
5
Prandtl number
Pr
5.12
The objectives of this work are summarized below: (a) Carry out time-step- and mesh-size-independent CFD simulations on confined flow occurring around a sphere of different diameters and fixed at different possible positions in a tube. (b) Study the effects of the particle eccentricity and blockage ratio on the drag and lift coefficient in the particle Reynolds number range of 0.1 e Rep e 500. Drag and lift coefficient correlations are proposed as functions of blockage ratio, eccentricity, and Rep. (c) Study the influence of blockage ratio and eccentricity on boundary layer separation from the sphere surface.
3. DEFINITIONS AND PROBLEM FORMULATION Incompressible laminar flow of a Newtonian liquid past a sphere placed in a tube is simulated numerically. The physical properties of the fluid are assumed constant and are given in Table 2. To prevent a possible loss of information due to the transition from two-dimensional (2-D) to three-dimensional (3-D) flow at different blockage ratios and eccentric positions, all the simulations were carried out in three dimensions. 3(a). Blockage Ratio. Different diameters of the sphere were used in the simulation while the tube diameter was maintained constant (at 50 mm). The ratio of the sphere diameter (Dp) to that of the tube diameter (DT) is defined as the blockage ratio (BR): BR ¼
Dp DT
ð1Þ
Blockage ratios in the range of BR = 0.020.5 were investigated. Results from BR = 0.02 correspond to the classical results. This low blockage ratio implies that a very small sphere has been 13139
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From eq 2, the above equation may be expressed as
Table 3. Range of Parameters Studies in This Work eccentricity, E
particle Reynolds number, maximum DF Vch F blockage ratio, Rep ¼ μ BR
Vch
0 e Rep e 500 0 e Rep e 500
0.5 0.5
Vch = Vmax Vch = 0.96Vmax
0.2
0 e Rep e 500
0.4
Vch = 0.84Vmax
0.3
0 e Rep e 500
0.2
Vch = 0.64Vmax
positioned centrally inside the tube and it is exposed to a flat (i.e., uniform) velocity profile (the classical case). 3(b). Eccentricity. For a sphere positioned at an eccentric distance (Ed) below the tube axis, the eccentricity (E) is expressed in terms of the tube diameter as follows: Ed ð2Þ DT 9,33 This definition is consistent with the previous work. The ranges of values for the various parameters considered in the simulations are listed in Table 3. 3(c). Particle Reynolds Number (Rep). The particle Reynolds number (Rep) is defined as follows: E¼
Dp Vch F μ
ð5Þ
velocity,
0 0.1
Rep ¼
Vch ¼ Vmax ð1 4E2 Þ
characteristic
ð3Þ
The fluid is assumed to have a well-developed parabolic velocity profile as it enters the tube and approaches the sphere. The characteristic velocity (Vch) is the value taken at that radial position of the incoming fluid velocity profile whose streamline terminates at the usual front stagnation point of the sphere. For a centrally positioned sphere, Vch will correspond to the maximum velocity of the incoming fluid. In the case of laminar flow of the entering fluid, the sphere, depending on its diameter, is exposed only to a part of the parabolic velocity profile of the fluid. This is in contrast to classical situations where the fluid velocity approaching the confined sphere is uniform. When the sphere is positioned at the axis of the tube, the characteristic velocity is equal to the maximum velocity of the approaching fluid. When a very small sphere is positioned centrally in a tube (i.e., the blockage ratio is small), it is exposed to essentially fluid velocities that are very close to that of the maximum fluid velocity. Even for reasonable blockage ratios (BR < 0.3), the sphere is exposed to a velocity profile closer to the maximum velocity rather than the average velocity. Hence, for the sake of consistency, the maximum velocity is used to define Rep when the particle is positioned at the center of the sphere. Furthermore, the simple relation of the maximum velocity being twice the average velocity in laminar tube flow enables correlations developed with the maximum-velocitybased Reynolds number to be simply adapted to cases where the average velocity is used to define the Reynolds number. When the sphere is positioned eccentrically, the incoming fluid velocity corresponding with the front stagnation point of the sphere may be found from eq 4. Here, R is the radial distance of the sphere center from the tube axis and equals the eccentric distance of the sphere (Ed) while Rmax is the tube radius. " # R 2 Vch ¼ Vmax 1 ð4Þ Rmax
The assumption of laminar flow for the maximum particle Reynolds number of 500 investigated in this work was checked. A laminar boundary layer developed, which rapidly thickened as the separation point was reached. Seeley et al.,2 based on experiments, observed uniform laminar flow around a sphere for Rep values up to 2940. They found there were no turbulent disturbances either in the boundary layer or in the free stream. Bagchi et al.23 carried out direct numerical simulations on hydrodynamics and heat transfer over a sphere immersed in a uniform flow of air. Laminar flow was considered for Rep values up to 500. Furthermore, the presence of a solid wall in confined flows will further contribute toward flow stabilization.14 The ranges of blockage ratios, eccentricities, and Rep values investigated in this work are summarized in Table 3. As the eccentricity increased and the sphere moved closer to the wall, the upper limit of the blockage ratio that can be possibly accommodated without the sphere touching the wall declined. 3(d). Clearance (C) per Unit Diameter of Tube. Clearance is the distance between the south pole of the sphere and the lower tube wall. It is defined as follows: C 1 ¼ ð1 2E BRÞ DT 2
ð6Þ
The relationship between clearance C, blockage ratio BR, and eccentricity E is given below. The parameter Γ is used in the correlation developed for the drag coefficients (see section 5.8, presented later in this work) is defined as follows: C Γ ¼ 12 ¼ 2E + BR ð7Þ DT 3(e). Drag and Lift Coefficients. The drag and lift coefficients are defined as follows:
CD ¼
ÆFD æ ð1=2ÞFVch 2 As
ð8Þ
ZZ ÆFD æ ¼ ½
CL ¼
ðP + τw Þn ds3ex ÆFL æ ð1=2ÞFVch 2 As
ZZ ÆFL æ ¼ ½ ðP + τw Þn ds3ey
ð9Þ
ð10Þ
ð11Þ
Here, ÆFDæ and ÆFLæ are the tangential and normal forces, respectively. These are obtained by integrating the local viscous and pressure forces along the sphere surface. Here, As is the projected area of the sphere, s denotes the surface area of the sphere, and n is the outward unit normal to the surface of the sphere. The parameters ex and ey are the unit vector in the x- and y-coordinate directions, respectively, and τw is the wall shear stress. 13140
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3(f). Dimensionless Sphere Surface Pressure (CP). The dimensionless sphere surface pressure (CP) is defined by
Ps P¥ ð1=2ÞFVch 2
CP ¼
ð12Þ
P¥ is a reference pressure is so chosen such that Cp, at the front stagnation point, is unity.14 This method of scaling, with respect to the stagnation point, helped to compare pressure distributions around the centrally positioned and eccentrically positioned spheres on a uniform basis. 3(g). Dimensionless Local Friction Factor (f). The Darcy friction factor (f) is defined as follows:9 f ¼
4τw ð1=2ÞFVch 2
ð13Þ
The friction factor in the y-coordinate direction (fy) is defined as follows: fy ¼
4τwy ð1=2ÞFVch 2
ð14Þ
where τwy is the vertical component of wall shear stress could be obtained directly in the post-processing step by the ANSYS CFX software35 used in the present work. The details are given in section 4. 3(h). Normalized Actual Drag Force (Ndf). The drag force at a given eccentricity was normalized with respect to drag force value that was estimated for the centrally positioned sphere and is denoted by Ndf. This is used later in section 5.2. 3(i). Stokes’ Law Normalized Drag Force (G). The actual drag coefficients computed in confined flows at different eccentric positions were normalized in terms of the Stokes law values applicable in the creeping flow regime (CDS = 24/Rep). This approach has been earlier adopted in Wham et al.8 and is used in section 5.8 of this study. The Stokes’ Law normalized drag force (G) is given as G¼
CD CDS
ð15Þ
4. DETAILS ON NUMERICAL SIMULATIONS 4(a). Governing Equations and Numerical Solution Methodology. The numerical studies were carried out using the
commercial ANSYS CFX 11 CFD software,35 which is based on the finite-volume method. The governing equations solved are listed below. (i) Continuity equation ∂F + r 3 ðFVÞ ¼ 0 ∂t
ð16Þ
(ii) Momentum equation ∂ðFVÞ + r 3 ðFV X VÞ ¼ r 3 ðpδ + μðrV + ðrVÞt ÞÞ ∂t ð17Þ 4(b). Initial and Boundary Conditions. Initially, the fluid is considered to be at rest inside the tube. At the inlet, the fully
developed velocity profile in the axial direction is given according to eq 4. At the sphere surface and tube walls, no-slip boundary conditions are imposed. At the exit boundary, the outflow boundary condition option pertaining to zero-diffusion flux for all flow variables was used. The pressure at the outlet was specified to be zero.35 In converged solutions, the outflow mass rate was checked and found to be equal to the inflow mass rate. From the numerical solution of the governing transport equations with the associated initial and boundary conditions, information on the velocity and pressure fields were obtained in the problem domain. 4(c). Details on the Mesh Size. The 3D grid with nonuniform structured and unstructured meshes was generated for different blockage ratios using the commercial ICEM CFX 11.35 The computational domain was divided into upstream, sphere, and downstream subdomains. In all of the subdomains, finer hexahedral mesh was generated near the wall surfaces including the sphere surface, in order to resolve rapid changes in velocities and pressure. A fluid/fluid interface was generated between the upstream subdomain and the sphere subdomain. The interface between the sphere subdomain and the downstream subdomain was also defined as a fluid/fluid interface. In the upstream and downstream cylindrical subdomains, an O-Grid was generated in the radial direction. A nonuniform grid was generated along the axial direction with an exponential reduction factor of 1.2 from the fluid inlet to the upstream subdomain/sphere subdomain interface. A similar grid scheme was adopted in the downstream section of the tube from the tube exit to the sphere subdomain/downstream subdomain interface. Structured meshes involving hexahedral elements could be generated efficiently for a sphere positioned at the tube center, as shown in Figure 1. Unstructured meshes were used in the sphere subdomain when the sphere was kept at eccentric distances from the tube axis. However, structured meshes involving hexahedral elements could be generated in the region immediately adjacent to the sphere for capturing rapid velocity variation. This could be done even when the sphere was placed close to the bottom tube wall (Ed = 0.3DT), as shown in Figure 1. For an eccentrically located sphere, the hybrid mesh scheme involved structured meshes only in the surrounding region close to the sphere and the tube wall surfaces in order to resolve the velocity gradients accurately. Furthermore, it was also difficult to construct a structured mesh when the sphere was eccentrically positioned. In the centrally located sphere case, a full structured mesh arrangement was possible. However, as requested by the anonymous reviewer, the hybrid scheme was incorporated even for the centrally positioned sphere for two representative cases, and the results have been compared in section 5.1(f). 4(d). Solution Methodology. This numerical study has been carried out using the robust and bounded second-order backward Euler scheme. The high-resolution scheme option, which maintains the blend factor as close as possible to unity without violating the boundedness principle, was chosen.35 Convergence criterion was fixed at a residuals root-mean-square (rms) value of
