Wall Retardation Effects on Flow and Drag Phenomena of Confined

Article pubs.acs.org/IECR

Wall Retardation Effects on Flow and Drag Phenomena of Confined Spherical Particles in Shear-Thickening Fluids C. Rajasekhar Reddy and Nanda Kishore* Department of Chemical Engineering, Indian Institute of Technology Guwahati, Assam − 781039, India ABSTRACT: In this work, effects of the wall retardation, Reynolds number, and shear-thickening viscosity behavior of fluids on flow and drag phenomena of confined spherical particles are presented. The governing mass and momentum conservation equations are solved using computational fluid dynamics-based commercial software. The numerical solver is thoroughly validated by comparing present results with existing literature for the case of unconfined spheres in Newtonian and shearthickening fluids. Extensive new results were presented in the following range of conditions: Reynolds number, Re, 1−100; wall factor, λ, 2−5; and power-law index, n, 1−1.8. The wall factor (λ) is defined as the ratio between the tube diameter and the particle diameter. The severity of wall retardation effects increases with increasing power-law index. For fixed values of the Reynolds number, the recirculation wake length decreases with decreasing wall factor and/or increasing power-law index. For n = 1.8, the wall retardation effects are very strong so that for λ = 2, there is no recirculation wake behind confined sphere even at Re = 100. Furthermore, regardless of values of the Reynolds number, the total drag coefficient increases with increasing power-law index and/or decreasing wall factor. The effect of the Reynolds number on the ratio between pressure and friction drag coefficients decreases with increasing power-law index and/or increasing wall factor. Finally, on the basis of present numerical results, a correlation is developed for the total drag coefficient of confined spherical particles settling in shear-thickening fluids. thickening behavior of fluids.4,5 These shear-thickening fluids under normal conditions behave as slightly viscous fluids; however, under the influence of external force, the viscosity of these fluids increases tremendously. Further, the moment the external force was released, they return to their normal slightly viscous fluid behavior. The shear-thickening fluids are generally prepared by suspending nonagglomerating nanoparticles in liquids, and some examples include corn starch in water, titanium oxide−water suspension, china clay−water suspension, etc. Shear-thickening fluids are used as drilling fluids in the oil industry to protect a well from blowouts, used to strengthen body armor, etc. Furthermore, there are several analytical and seminumerical results on the settling velocity of unconfined spheres in powerlaw fluids in the creeping flow regime, which are thoroughly reviewed in a recent book.4 Perhaps, Tripathi and Chhabra6 are the first to report numerical results on the settling behavior of unconfined spheres and spheroids in shear-thickening fluids. Recently, Dhole et al.7,8 reported results on drag and heat transfer behavior of unconfined sphere in shear-thinning and shear-thickening fluids in the intermediate range of pertinent variables, while Song et al.9−11 studied the effects of moving wall on the momentum and heat transfer characteristics of confined spheres in shear-thinning fluids only. However, to the best of the authors’ knowledge, no results for wall retardation effects on confined spheres settling in shear-thickening fluids are available even in the creeping flow regime, let alone the

1. INTRODUCTION The settling velocity (or the drag coefficient) of solid particles in viscous fluids is a prerequisite to the design of solid−liquid contacting equipments in many process industries. If the design details of such contacting equipments are already known, then the information of drag coefficients of solid particles is useful in rationalizing solid−liquid equipments, or for the mechanical separation of different phases contacting in those equipments. In such processes, often one encounters a variety of irregular particles interacting with neighboring particles, wall, and surrounding fluids; however, the drag coefficients of isolated regular particles can provide adequate information concerned to the physics of such equipments. Hence, voluminous literature has been accrued on the flow and drag phenomena of regular particles such as spheres, cylinders, cubes, etc., settling in Newtonian1−3 and in a variety of non-Newtonian liquids.4 Furthermore, the information related to the settling velocity of bluff bodies can be conveniently presented in terms of nondimensional numbers such as the drag coefficients as functions of the Reynolds number, wall retardation factor, and the characteristic constants of the rheology of surrounding fluids. On the other hand, many fluids in chemical, pharmaceutical, food, polymer, and other processing industries display a wide range of non-Newtonian characteristics including shearthinning, shear-thickening, yield stress, and viscoelastic behavior. However, the majority of aforementioned industrial fluids display Ostwald−de Waele or power-law rheological characteristics including shear-thinning and shear-thickening nature. In the last couple of decades, enormous literature has been published related to shear-thinning fluids, while the shearthickening fluids are investigated to a lesser extent. However, with the growing importance of highly loaded process systems, there has been renewed interest in studying the shear© 2012 American Chemical Society

Received: Revised: Accepted: Published: 16755

October 4, 2012 November 30, 2012 December 3, 2012 December 3, 2012 dx.doi.org/10.1021/ie302707s | Ind. Eng. Chem. Res. 2012, 51, 16755−16762

Industrial & Engineering Chemistry Research

Article

0.1 ≤ n ≤ 1 and Re → 0. They compared their numerical solutions to the upper and lower bounds of theoretical solutions and experimental data. They have also provided some remarks on wall effects and observed that the wall effects are negligible for n ≤ 0.5. An approximate boundary layer solution to the momentum equations for power-law fluid flow past a solid sphere was found by Bizzell and Slattery23 in the range of 0.7356 ≤ n ≤ 1, and by Lin and Chern24 in the range of 0.2 ≤ n ≤ 1. Tripathi et al.25 obtained numerical solution of psuedoplastic or shear-thinning fluid (n < 1) flow past unconfined spherical and spheroidal particles at intermediate range of Reynolds number up to 100 and 0.4 ≤ n ≤ 1 using a finite element method-based solver. Later, Tripathi and Chhabra6 extended this study for shear-thickening fluids (n > 1) in the range of conditions 0.001 ≤ Re ≤ 100 and 1 ≤ n ≤ 1.8. However, their results are found to be inaccurate for power-law fluids.7 Graham and Jones26 numerically investigated the settling of spherical particles in power-law fluids in the range of 0.2 ≤ Re ≤ 100 and 0.4 ≤ n ≤ 1. They used a commercial package POLYFLOW, which is based on a finite element method and proposed two correlations for the estimation of drag on a sphere in power-law liquids for two different ranges of the Reynolds number. Ceylan et al.27 developed a theoretical approach for the estimation of drag force using the previously available correction factors to Stokes’ drag as applied to the power-law fluids and developed a correlation that is in good agreement with the experimental results in the range of 10−3 ≤ Re ≤ 103 and 0.5 ≤ n ≤ 1. Recently, Dhole et al.7 numerically obtained the drag coefficients of a power-law fluid flow past a single unconfined solid sphere in the range of conditions 5 ≤ Re ≤ 200 and 0.5 ≤ n ≤ 1.8. They used a Cartesian coordinate system-based finite volume solver and a sphere-in-a-tube computational domain. However, they have taken a large value of tube to particle diameter ratio so as to ensure the so-called unbounded flow conditions prevail, and they proposed a correlation for the drag coefficient of unconfined sphere settling in power-law fluids. Finally, on the basis of analytical, theoretical, experimental, and numerical studies, reliable information is now available on the drag curve of unconfined spherical particles freely settling in power-law fluids (both shear-thinning and shear-thickening) at least in the steady axisymmetric flow regime. On the other hand, for the case of confined spheres, literature is confined to the case of Newtonian (n = 1) and shear-thinning fluids (n < 1) only. Some of the key mentioning of such studies is included herein. Kawase and Ulbrecht28 found that the shear-thinning behavior suppresses the wall effects in the creeping flow regime. Missirlis et al.29 considered the sphere falling axially in an infinitely long cylinder filled with shear-thinning type power-law fluids in the creeping flow regime over the conditions n = 0.1−1 and λ = 2−50. They also found that the wall effect suppressed with decreasing power-law index, which is also consistent with Hsu et al.30 Recently, Song et al.9 numerically investigated the confined flow of a sphere in shear-thinning fluids (n < 1) over intermediate range of conditions and concluded that the effect of the confining wall is severe at low Reynolds numbers. However, to the best of the authors’ knowledge, there is a large gap in the flow and drag phenomena of confined spheres in shear-thickening fluids (n > 1) that even in the limit of creeping flow regime no results are available. Thus, the aim of this work is to elucidate effects of pertinent dimensionless parameters on the flow and drag

intermediate to large Reynolds numbers regime. Thus, the aim of this work is to fill this gap in the literature.

2. PREVIOUS WORK The flow of a viscous fluid past a stationary isolated sphere may be considered as a simplified case of a general family of immersed bluff-body flows with widespread applications. The behavior of the flow past a sphere at varying Reynolds number has been studied by a number of researchers. The steady motion of a solid sphere in a viscous fluid had been a subject of main interest from the early 18th century when Newton did some experiments to calculate the drag force on a solid sphere in 1710.2 Newton obtained an expression for the drag force acting on a solid sphere as Fd = 0.055πd2ρV2, which mainly accounts for inertial effects. At very low velocities, Stokes12 suggested that the inertial effects are so small that they can be neglected from the momentum equations and obtained a resulting drag force as Fd = 3πμdV. Subsequent experiments in the intermediate range of Reynolds number between Stokes’ and Newton’s relation yielded the so-called drag curve applicable to a solid sphere settling in a quiescent infinite expanse of Newtonian fluid.1−3 An analogous drag curve of an unconfined sphere in power-law liquids is also established at least in the steady axisymmetric flow regime.4 However, a recount of some of this literature is important for the completeness of this section. Until the recent past, the majority of the theoretical studies of non-Newtonian flow past a solid sphere are applicable to the creeping flow only and are thoroughly reviewed by Chhabra.4 A brief recount of these studies reveals that most of the investigators studied power-law model fluids,13−15 but considerable works have also been done with Ellis model fluids and Carreau viscosity model fluids.14 Because of the nonlinearity of the viscous terms (because of non-Newtonian viscosity), governing momentum equations are highly nonlinear even when the inertial effects are neglected. Therefore, an analytical closed form solution, akin to the Stokes’ expression, is not possible even for the simple power-law model. Consequently, only approximate solutions are available even in the creeping flow regime. These approximate solutions are majorly based on the use of velocity and stress variational principles.16−18 These approximate solutions yield upper and lower bounds on the drag force, and in the absence of any definitive information, the use of the arithmetic mean of the two bounds is suggested. These individual or the mean of upper and lower bounds can thus be expressed as the correction factor, Y, applied to the Stokes’ expression, defined as Cd = (24Y/Re). For instance, Lockyer et al.19 have experimentally obtained this correction factor as (2 − n) in the range of Re ≤ 0.2. Adachi et al.20 seem to be the first to report the numerical solution of the Newtonian (n = 1) and shear-thinning type power-law fluids (n < 1) flow past an unconfined solid sphere at a single value of Reynolds number of 60 and power-law index between 1 and 0.8. Adachi et al.20 solved the spherical coordinate system-based stream functions and vorticity transport equations by using an extended moments method. They observed that as the powerlaw index decreases, the friction drag decreases, whereas the pressure and total drag increase at Re = 60. Bush and PhanThein21 used the boundary element method to obtain numerical solution of a Carreau fluid flow past a solid sphere. Dhazi and Tanner22 numerically obtained the drag on an unbounded solid sphere in an inelastic power-law fluid by using a finite element method-based solver in the range of conditions 16756

dx.doi.org/10.1021/ie302707s | Ind. Eng. Chem. Res. 2012, 51, 16755−16762


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Figure 1. Schematic representation of flow past a confined sphere.

The appropriate boundary conditions for this flow problem can be ascribed as follows. • At the inlet:

phenomena of confined spheres in shear-thickening fluids in the range of conditions: Re = 1−100, λ = 2−5, and n = 1−1.8.

3. PROBLEM STATEMENT AND MATHEMATICAL FORMULATION As shown in Figure 1, consider a spherical particle along the central axis of a long cylindrical tube, and assume that the particle is fixed at an upstream distance of Lu from the inlet and a downstream distance of Ld from the exit of the tube. The length of the cylindrical tube is L, and the diameter is Dt. Further, the length of the tube is taken very long so that the entry and exit effects are negligible. An incompressible fluid is entering the tube at a velocity of V and pressure Po at the inlet and leaves at a pressure P from the outlet of the tube. In the numerical solver, the wall factor, to account for the effect of wall retardation, is imposed by altering the diameter of the tube. It is defined as the ratio between diameters of the cylindrical tube and the particle. Furthermore, the flow is assumed to be two-dimensional, steady, and axisymmetric. In the present range of conditions, these assumptions are consistent with other relevant numerical studies in the literature.6−11,26,30 The governing equations for this fluid flow problem are the conservation of mass and momentum as written: • Continuity equation: ∇·V = 0

Vx = V ; Vy = 0

• Along the surface of sphere: Vx = 0; Vy = 0

Vx = 0; Vy = 0

∂Vx = 0; Vy = 0 ∂y

(8)

The governing conservation equations of mass and momentum along with aforementioned boundary conditions are solved using commercial software based on computational fluid dynamics, Fluent 6.3, along with mesh generating software, Gambit. The Fluent software is based on the finite volume method, and it utilizes the conservation equations and boundary conditions in the Cartesian coordinates to obtain solution for flow past any complex geometry. Once the fully converged velocity and pressure fields are obtained, then these are further used to evaluate the streamlines and drag coefficients, and are reported as functions of pertinent dimensionless numbers. The dimensionless numbers used in this work are defined below.

(1)

(2)

• Reynolds number:

(3)

where ρ and η are the density and viscosity of the fluid, respectively. Furthermore, the viscosity equation for a powerlaw fluid can be written as: ⎛ I ⎞(n − 1)/2 η = m⎜ 2 ⎟ ⎝2⎠

(7)

• Along the central axis of the tube:

The extra stress tensor for an incompressible fluid is defined as: τ = 2ηε

(6)

• Along the tube wall:

• Momentum equation: ρ[(V ·∇)V ] = −∇P + ∇·τ

(5)

Re =

d nV 2 − nρ m

(9)

• Wall factor: (4)

λ=

where m and n are characteristics of a power-law fluid, referred to as power-law consistency index and power-law behavior index, respectively. The rate of deformation tensor (ε) and second invariant of the rate of deformation tensor (I2) are related to velocity components and their derivatives, which can be found in any standard transport phenomena book (for instance, see Bird et al.31). From the power-law viscosity model, one can recover the Newtonian behavior by substituting n = 1. If n < 1, the fluid is known to display shear-thinning behavior, and for n > 1, the fluid is referred to as shear-thickening fluid.

Dt d

(10)

• Total drag coefficient: Cd =

Fd 2

( 12 ρV 2)( π4d )

= Cdp + Cdf (11)

where Fd is the drag force experienced by the particle, and Cdp and Cdf are pressure and viscous components of the total drag coefficient, respectively. 16757



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4. NUMERICAL METHODOLOGY Because the detailed numerical procedure to obtain the steady velocity and pressure distributions in the entire computational domain is available elsewhere,32,33 only salient features are presented herein. The conservation of mass and momentum equations are solved using the semi-implicit method for pressure-linked equations (SIMPLE) algorithm along with the quadratic interpolation for the convective kinematics (QUICK) scheme for convective terms. The computations to obtain steady velocity and pressure fields are carried out until the residual values of the continuity, x-, and y-components of momentum equations decrease to 10−8. The fully converged velocity and pressure fields are further used to evaluate the near-particle kinematics such as the drag coefficients and streamlines. However, before presenting new results, it is mandatory to check the numerical artifacts such as entry/exit effects and grid independence of the numerical solver. 4.1. Entry and Exit Effects. In confined flow problems, one of the dimensions (the diameter of the cylindrical tube) of the computational domain size is fixed depending on the value of the wall factor or the confinement ratio; however, the other dimensions such as the length of the tube, upstream, and downstream distances should be found through numerical experiments so that the entry and exit effects are negligible. For this purpose, different values of Lu and Ld are examined for extreme values of the power-law index and the wall factor at Re = 1 and are presented in Table 1. Although all values of upstream and downstream distances produce almost identical results, larger values of Lu and Ld, that is, Lu = Ld = 65, are chosen.

Table 2. Effect of Grid on Total Drag Coefficients of Confined Spherical Particles at Re = 100

Lu

Ld

n=1

n = 1.8

n=1

n = 1.8

35 50 65

240.28 240.15 239.45

1344.6 1344.0 1341.0

79.38 79.48 79.42

150.61 151.01 150.96

Grid-1 Grid-2 Grid-3

100 150 200

300 350 400

Grid-1 Grid-2 Grid-3

100 150 200

300 350 400

Ld

tube wall

n=1

n = 1.8

850 900 950

3.80 3.82 3.82

14.82 14.93 14.93

850 900 950

2.67 2.67 2.67

6.09 6.11 6.11

λ=2 300 350 400 λ=5 300 350 400

Table 3. Comparison of Total Drag Coefficient of Unconfined Spheres in Newtonian Fluids Re

Le Clair et al.34

Saboni et al.35

Kishore et al.36

Dhole et al.7

present

1 20 200

27.37 2.736 0.772

27.55 2.768 0.776

27.708 2.739 0.779

26.13 2.692 0.744

27.457 2.726 0.775

unconfined sphere in Newtonian fluids. It is clear from this table that the agreement with literature values obtained by various numerical methods including primitive variables (both finite difference and finite volume solvers) and stream function−vorticity approaches is excellent. Table 4 shows a

λ=5

35 50 65

Lu

sphere surface

5. RESULTS AND DISCUSSION To delineate effects of the wall confinement on the flow and drag characteristics of confined spheres, the following range of conditions are considered: Re = 1, 5, 10, 20, 50, 100, λ = 2, 3, 5, and n = 1, 1.2, 1.4, 1.6, 1.8. Before new results are presented, however, it is customary to validate the accuracy and reliability of the present numerical solver. 5.1. Validation. Table 3 presents a comparison of present drag coefficients with literature values for the case of

Table 1. Entry and Exit Effects on Total Drag Coefficients of Confined Spheres at Re = 1 λ=2

Cd

no. of nodes grid

Table 4. Comparison of Drag Coefficient of Confined Spheres in Newtonian Fluids λ=3

4.2. Grid Independence Study. In such bluff body flow systems, the effect of the grid is usually very significant along the surface where fluid is in contact with the solid object because of the possibility of sharp velocity gradients with increasing Reynolds numbers. In the case of unconfined flow past a single particle, the velocity gradient is strong along the surface of the particle, and it gradually decays as one moves away from the particle, and then gradients vanish when at a far distance from the particle. However, in the case of confined flow past a particle, the gradients of velocity will be higher at the particle surface as well as along the tube wall, which is in close proximity to the particle. Therefore, it is mandatory to have a finer grid both on the surface of particle and along the tube. Further, the clustering of grid around the sphere can be obtained by choosing a finer grid along the central axis with a specified aspect ratio. Considering these issues, a thorough grid independence study is carried out and presented in Table 2. Although all grids produce almost identical results, for all combinations of Reynolds number and power-law index, Grid-3 is chosen for λ = 2, and Grid-2 is used for λ = 3 and λ = 5.

λ=5

Re

Wham et al.37

Kishore and Gu32

present

Wham et al.37

Kishore and Gu32

present

1 10 20 50

126.934 14.235 8.252 4.574

120.681 13.648 7.959 4.374

120.431 13.486 7.856 4.338

81.854 11.412 7.266 4.349

79.775 11.031 6.938 3.965

79.467 11.001 7.019 4.101

comparison between present drag coefficients of confined spheres in Newtonian liquids with literature values.32,37 From Table 4, it can be seen that as compared to Wham et al.,37 the present results are within ±5.7% (maximum difference for λ = 5 at Re = 50), while as compared to Kishore and Gu,32 the present results are within ±3.4% (maximum difference for λ = 5 at Re = 50) absolute relative difference. Table 5 shows a comparison between present values of drag coefficients of unconfined spheres in shear-thickening fluids with those of Dhole et al.7 obtained by the SIMPLE algorithm implemented by using a finite volume approach, and the agreement between the two results is good and the maximum deviation is found to be less than ±5% in the case of Re = 20 and n = 1.8. Finally, a 16758



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2c). On the other hand, at Re = 100, for all values of the wall factor, a recirculation wake is observed in the rear end, indicating the effect of the presence of solid wall is relatively small in comparison to small values of the Reynolds number. This is because at large Reynolds numbers, the contribution of the convection force is large as compared to viscous force. By decreasing the wall confinement, that is, by increasing the value of λ, the length of recirculation wake increases; however, it is smaller than the case of unconfined sphere. Figure 3 shows

Table 5. Comparison of Drag Coefficient of Unconfined Spheres in Shear-Thickening Fluids Re = 5 n

Dhole et al.7

1.2 1.4 1.6 1.8

7.08 7.02 6.93 6.83

Re = 20 present

Dhole et al.7

7.03 6.88 6.75 6.58

2.96 3.15 3.28 3.41

Re = 100

present

Dhole et al.7

present

2.89 3.02 3.14 3.24

1.26 1.41 1.57 1.69

1.24 1.37 1.50 1.62

comparison on drag coefficients of confined spheres (λ = 2) in shear-thinning fluids to those of Song et al.9 is presented in Table 6 by adopting moving wall boundary condition along the Table 6. Comparison of Drag Coefficient of Confined Spheres (λ = 2) in Shear-Thinning Fluids with Moving Wall Boundary Condition n = 0.8

n = 0.6

Re

Song et al.9

present

Song et al.9

present

1 5 10 50 100

103.7 20.83 11.02 2.854 1.829

104.9 21.13 10.78 2.797 1.820

77.23 15.84 8.139 2.163 1.441

76.97 15.57 8.013 2.140 1.409

Figure 3. Streamlines of shear-thickening fluid (n = 1.4) flow past confined sphere at Re = 100 (upper half) and Re = 20 (lower half).

tube wall, and the present results are within ±2.2% of absolute relative difference as compared to Song et al.9 Furthermore, discrepancies in such numerical studies are not at all unknown and are often ascribed to the differences in numerical approach, grid resolution, etc.38,39 Therefore, on the basis of aforementioned rigorous validation, it is safe to conclude that the present solver is reliable and accurate to within ±4−5%. 5.2. Flow Phenomena. Figure 2 shows the streamline patterns of Newtonian flow past a confined sphere at Re = 100

streamline patters of a shear-thickening fluid (n = 1.4) flow past a confined sphere at Re = 100 (upper half) and Re = 20 (lower half). For both values of the Reynolds number, the trends are qualitatively similar to the case of Newtonian fluids; however, the lengths of the recirculation wakes are clearly affected because of the shear-thickening behavior of the fluid. For shearthickening fluids, as the rate of shear increases, the apparent viscosity also increases, and thus the rate of settling of the sphere will be lowered as compared to Newtonian fluids. In other words, along with the wall retardation effect, the nonNewtonian behavior top up the viscous forces; hence, for fixed values of the Reynolds number and the wall factor, the length of the recirculation wake behind the sphere is reduced as compared to the case of Newtonian fluid. Therefore, for all values of the Reynolds number and the wall factor, the length of recirculation wake for a shear-thickening fluid of n = 1.4 is reduced as compared to the case of Newtonian fluids. Figure 4 represents the streamline patterns of a shear-thickening fluid (n = 1.8) flow past a confined sphere at Re = 100 (upper half of the central line) and Re = 20 (lower half of the central line) for different values of the wall factor; and qualitatively similar trends can be seen as in the case of Figures 2 and 3. However, for a given combination of the Reynolds number and wall factor, the lengths of recirculation wakes are substantially smaller as compared to the case of the Newtonian fluids (see Figure 2) because of the aforementioned reasons. In the case of n = 1.8, the addendum to viscous effects because of contribution from the wall retardation and shear-thickening behavior of fluid is very high that for λ = 2, there is no recirculation wake even at Re = 100. This signifies the degree of suppress of convection forces because of the presence of a wall near to the particle and the shear-thickening viscosity behavior of the fluid. In summary, the length of the recirculation wake behind confined spheres decreases with increasing power-law

Figure 2. Streamlines of Newtonian fluid (n = 1) flow past confined sphere at Re = 100 (upper half) and Re = 20 (lower half).

(upper half of the central line) and Re = 20 (lower half of the central line) for different values of the wall factor. From Figure 2a, at Re = 20 and for λ = 2, almost fore-aft symmetry of streamlines was observed because the presence of wall suppressed the convection force to a greater extent, and hence viscous forces are expected to dominate. However, as the value of the wall factor increased to 3 and further to 5, the retardation effect of the wall is reduced, and a small recirculation wake is observed in the rear of the sphere (Figure 16759



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to 5. This can be attributed to decreasing confinement effect with increasing values of the wall factor. On the other hand, the relative contribution of pressure and friction drag coefficients is also germane to the proper rationalizing of the solid−liquid contactors. Hence, Figure 6 represents the combined effects of the Reynolds number, power-law index, and the wall factor on the relative contributions of pressure and friction drag coefficients. For the case of Newtonian fluids, for all values of the wall factor, as the value of the Reynolds number increases, the ratio between two individual drag coefficients gradually increases. However, as the value of the power-law index increased gradually from n = 1 to n = 1.8, the effect of Reynolds number becomes very small (that can be neglected) for λ = 2 and λ = 3, although a small but finite effect is found for λ = 5. The reason for this behavior can be ascribed to the increasing retardation effect for small values of the wall factor and/or for large values of the power-law index. In summary, the total drag coefficient decreases with increasing Reynolds number and/or wall factor, and/or decreasing values of the power-law index. The effect of the Reynolds number on relative contribution of pressure and friction drag coefficients is large for Newtonian fluids, and it decreases for shear-thickening fluids. Finally, on the basis of the present numerical results, the following form for the total drag coefficient is proposed, which reproduces the present results with an average relative difference of ±8.88% that rises to a maximum of ±28.6% for extreme cases.

Figure 4. Streamlines of shear-thickening fluid (n = 1.8) flow past confined sphere at Re = 100 (upper half) and Re = 20 (lower half).

index and/or wall factor, and/or with decreasing Reynolds number. 5.3. Drag Phenomena. In any solid−liquid contacting equipments, while settling, solid particles experience a variety of forces including hydrodynamic forces, out of which the drag force experienced by the solid particle is of prime importance in the design of such contactors. Furthermore, in the majority of such equipment, the fluid phase displays a variety of nonNewtonian characteristics including the shear-thickening behavior. Therefore, this subsection is devoted to the drag behavior of confined spherical particles in shear-thickening fluids over a wide range of present conditions. Figure 5 shows the total drag coefficients of confined sphere in shearthickening fluids as functions of the Reynolds number and the wall factor. For all values of the wall factor and the powerlaw index, the characteristic drag behavior is observed; that is, as the value of the Reynolds number increases, the drag coefficient decreases. For fixed values of the Reynolds number and the power-law index, the drag coefficient decreases with increasing wall factor. For fixed values of the Reynolds number and the wall factor, the drag values increases with increasing power-law index; that is, for a given combination of Re and λ, the drag coefficients of shear-thickening fluids are larger than those of Newtonian fluids. However, the effect of power-law index is more significant for small values of the wall factor because of strong retardation effects of the wall. As the value of the wall factor increases from 2 to 3, the difference in drag coefficient values with increasing power-law index is small, which further decreases as the value of the wall factor increases

Cd =

2(2n + 1)(n + 2) (1 + 0.157Re 0.687) Re ⎡ ⎤3.1 n−0.005Re + 0.6Re−0.16 ⎢1 + ⎥ ⎣ (0.001Re + 1.63n0.88 − 0.55Re 0.16(λ − 2)(λ − 3))λ − 2 ⎦

(12)

The above correlation reproduces the present numerical results within the following range of conditions: 1 ≤ Re ≤ 100, 1 ≤ n ≤ 1.8, and 2 ≤ λ ≤ 5. Further, eq 12 reduces to the well-known correlation of Schiller and Naumann40 for total drag coefficients of solid particles settling in unbounded (large λ) Newtonian fluids (n = 1). Figure 7 shows a parity plot between present numerical drag values and those obtained by above correlation, and there are no discernible trends for any combination of pertinent parameters.

6. CONCLUSIONS The flow and drag phenomena of confined spherical particles in shear-thickening fluids are thoroughly investigated through numerical approach for a wide range of conditions. The length of the recirculation wake behind confined spheres decreases

Figure 5. Total drag coefficient of confined spherical particles in shear-thickening fluids. 16760



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Figure 6. Ratio between pressure and friction drag coefficients of confined spherical particles in shear-thickening fluids.

V = velocity vector, m/s Vx = x-component of velocity, m/s Vy = y-component of velocity, m/s Greek Symbols

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REFERENCES

(1) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic Press: New York, 1978. (2) Soo, S. L. Multiphase Fluid Dynamics; Science Press: Beijing, China, 1990. (3) Michaelides, E. E. Particles, Bubbles and Drops: Their Motion, Heat and Mass Transfer; World Scientific: Singapore, 2006. (4) Chhabra, R. P. Bubbles, Drops and Particles in Non-Newtonian Fluids; CRC Press: Boca Raton, FL, 2006. (5) Chhabra, R. P.; Richardson, J. F. Non-Newtonian Flow and Applied Rheology; Butterworth−Heinemann: Oxford, U.K., 2008. (6) Tripathi, A.; Chhabra, R. P. Drag on spheroidal particles in dilatant fluids. AIChE J. 1995, 41, 728−731. (7) Dhole, S. D.; Chhabra, R. P.; Eswaran, V. Flow of power-law fluids past a sphere at intermediate Reynolds numbers. Ind. Eng. Chem. Res. 2006, 45, 4773−4781. (8) Dhole, S. D.; Chhabra, R. P.; Eswaran, V. Forced convection heat transfer from a sphere to non-Newtonian power-law fluids. AIChE J. 2006, 52, 3658−3667. (9) Song, D.; Gupta, R. K.; Chhabra, R. P. Wall effects on a sphere falling in quiescent power law fluids in cylindrical tubes. Ind. Eng. Chem. Res. 2009, 48, 5845−5856. (10) Song, D.; Gupta, R. K.; Chhabra, R. P. Effect of blockage on heat transfer from a sphere in power-law fluids. Ind. Eng. Chem. Res. 2010, 49, 3849−3861. (11) Song, D.; Gupta, R. K.; Chhabra, R. P. Heat transfer to a sphere in tube flow of power-law liquids. Int. J. Heat Mass Transfer 2012, 55, 2110−2121. (12) Stokes, G. G. On the effect of internal friction of fluids on Pendulums. Trans. Cambridge Philos. Soc. 1851, 9, 8. (13) Acharya, A.; Mashelkar, R. A.; Ulbrecht, J. Flow of inelastic and viscoelastic fluid past a sphere. Rheol. Acta 1976, 15, 454−470. (14) Chhabra, R. P.; Uhlherr, P. H. T. Sphere motion through nonNewtonian fluids at high Reynolds numbers. Can. J. Chem. Eng. 1980, 58, 124−128. (15) Lali, A. M.; Khare, A. S.; Joshi, J. B.; Nigam, K. D. P. Behaviour of solid particles in viscous non-Newtonian solutions: settling velocity, wall effects and bed expansion in solid-liquid fluidized beds. Powder Technol. 1989, 57, 39−50. (16) Wasserman, M. L.; Slattery, J. C. Upper and lower bounds on the drag coefficient of a sphere in a power-model. AIChE J. 1964, 10, 383−388. (17) Cho, Y. I.; Hartnett, J. P. Drag coefficients of a slowly moving sphere in non-Newtonian fluids. J. Non-Newtonian Fluid Mech. 1983, 12, 243−247.

Figure 7. Parity plot between present numerical and correlated values of Cd.

with increasing values of the power-law index and/or the wall factor, and/or with decreasing value of the Reynolds number. For λ = 2 and n = 1.8, there is no recirculation wake behind the confined sphere even at Re = 100. The total drag coefficient decreases with increasing Reynolds number and/or the wall factor, and/or decreasing values of the power-law index. The effect of the Reynolds number on relative contribution of pressure and friction drag coefficients is large for Newtonian fluids, and it decreases for shear-thickening fluids. Finally, on the basis of the present numerical results, a correlation is proposed for the total drag coefficient of a confined sphere settling in shear-thickening fluids.

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λ = wall factor, dimensionless η = viscosity of fluid, Pa s ρ = density of fluid, kg/m3

AUTHOR INFORMATION

Corresponding Author

*Tel.: +91-361-2582276. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOMENCLATURE Cd = total drag coefficient, dimensionless Cdf = friction drag coefficient, dimensionless Cdp = pressure drag coefficient, dimensionless d = diameter of sphere, m Dt = diameter of the tube, m Fd = drag force, N L = length of the tube, m Ld = downstream distance, m Lu = upstream distance, m m = power-law fluid viscosity consistency index, Pa sn n = power-law fluid viscosity behavior index, dimensionless P = pressure, Pa Re = Reynolds number, dimensionless 16761



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(18) Kawase, Y.; Moo-Young, M. Approximate solutions for powerlaw fluid flow past a particle at low Reynolds number. J. NonNewtonian Fluid Mech. 1986, 21, 167−177. (19) Lockyer, M. A.; Davies, A. R.; Jones, T. E. R. The importance of rheology in the determination of the carrying capacity of oil-drilling fluids. Int. Conf. Plastics Med. Surgery 1980, 2, 127−132. (20) Adachi, K.; Yoshioka, N.; Yamamoto, K. On non-Newtonian flow past a sphere. Chem. Eng. Sci. 1973, 28, 2033−2043. (21) Bush, M. B.; Phan-Thien, N. Drag force on a sphere in creeping motion through a Carreau model fluid. J. Non-Newtonian Fluid Mech. 1984, 16, 303−313. (22) Dhazi, G.; Tanner, R. I. The drag on a sphere in power-law fluid. J. Non-Newtonian Fluid Mech. 1985, 17, 1−12. (23) Bizzell, G. D.; Slattery, J. C. Non-Newtonian boundary-layer flow. Chem. Eng. Sci. 1962, 17, 777−782. (24) Lin, F. N.; Chern, S. Y. Laminar boundary layer flow of nonNewtonian fluid. Int. J. Heat Mass Transfer 1979, 22, 1323−1329. (25) Tripathi, A.; Chhabra, R. P.; Sundararajan, T. Power-law fluid flow over spheroidal particles. Ind. Eng. Chem. Res. 1994, 33, 403−410. (26) Graham, D. I.; Jones, T. E. R. Settling and transport of spherical particles in power-law fluids at finite Reynolds number. J. NonNewtonian Fluid Mech. 1994, 54, 465−488. (27) Ceylan, K.; Herdem, S.; Abbasov, T. A theoreticl model for estimation of drag force in the flow of non-Newtonian fluids around spherical solid particles. Powder Technol. 1999, 103, 286−291. (28) Kawase, Y.; Ulbrecht, J. J. The influence of walls on the motion of a sphere in non-Newtonian fluids. Rheol. Acta 1983, 22, 27−34. (29) Missirlis, K. A.; Assimacopoulos, D.; Mitsoulis, E.; Chhabra, R. P. Wall effects for motion of spheres in shear-thinning fluids. J. NonNewtonian Fluid Mech. 2001, 96, 459−471. (30) Hsu, J. P.; Hsieh, Y.; Tseng, S. Drag force on rigid spheroidal particles in a cylinder filled with Carreu fluid. J. Colloid Interface Sci. 2005, 284, 729−741. (31) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley: New York, 2002. (32) Kishore, N.; Gu, S. Wall effects on flow and drag phenomena of spheroid particles at moderate Reynolds numbers. Ind. Eng. Chem. Res. 2010, 49, 9486−9495. (33) Kishore, N.; Gu, S. Momentum and heat transfer phenomena of spheroid particles at moderate Reynolds and Prandtl numbers. Int. J. Heat Mass Transfer 2011, 54, 2595−2601. (34) LeClair, B. P.; Hamielec, A. E.; Pruppacher, H. R. A numerical study of the drag on a sphere at intermediate Reynolds numbers. J. Atoms Sci. 1970, 27, 308−315. (35) Saboni, A.; Alexandrova, S.; Gourdon, C. Determination de la trainee engendree par une sphere fluide en translation. Chem. Eng. J. 2004, 98, 175−182. (36) Kishore, N.; Chhabra, R. P.; Eswaran, V. Drag on a single fluid sphere translating in power-law liquids at moderate Reynolds numbers. Chem. Eng. Sci. 2007, 62, 2422−2434. (37) Wham, R. M.; Basaran, O. A.; Byers, C. H. Wall effects on flow past solid spheres at finite Reynolds numbers. Ind. Eng. Chem. Res. 1996, 35, 864−874. (38) Roache, P. J. Perspective: A method for uniform reporting of grid refinement studies. Trans. ASME, J. Fluids Eng. 1994, 116, 405− 413. (39) Zimmerman, W. B. Multiphysics Modelling with Finite Element Methods; World Scientific: Singapore, 2006. (40) Schiller, L.; Naumann, A. Z. Uber die grundlegenden Berechungen bei der Schwerkraflaufbereitung. Ver. Dtsch. Ing. 1933, 77, 318−320.

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Wall Retardation Effects on Flow and Drag Phenomena of Confined

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