Sedimentation of a cylindrical particle in a Carreau fluid

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Sedimentation of a cylindrical particle in a Carreau fluid

Jyh-Ping Hsu

a,∗

_{, Ching-Feng Shie}

a

_{, Shiojenn Tseng}

b

a_{Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan} b_{Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan}

Received 11 October 2004; accepted 19 January 2005 Available online 10 March 2005

Abstract

The drag coefficient of an isolated, rigid cylindrical particle in a Carreau fluid is evaluated. The result of numerical simulation reveals that, in general, the shear-thinning nature of a Carreau fluid yields a drag coefficient smaller than that for the corresponding Newtonian fluid. Also, the smaller the Reynolds number, the more appreciable the decrease of the drag coefficient as the relaxation time constant of the Carreau fluid increases. The influence of the index parameter of a Carreau fluid on the drag coefficient depends largely on the magnitude of the relaxation time constant and is insensitive to the Reynolds number. Only if the relaxation time constant is sufficiently large is the influence of the index parameter on the drag coefficient significant. If the Reynolds number and/or the relaxation time constant is sufficiently large, the flow field upstream of a particle becomes asymmetric to that downstream. In general, the influence of the index parameter, the relaxation time constant, and the Reynolds number on the flow field follows the order index parameter < relaxation time constant < Reynolds number.

Keywords: Sedimentation; Cylindrical particle; Carreau fluid; Drag coefficient

1. Introduction

The terminal velocity of particles in a fluid medium is one of its basic characteristics. Through appropriate measure-ments, the drag acting on a particle can be estimated, which, in turn, can be used to evaluate its physical properties. Practi-cal application includes, for example, design of viscometers, fluidized beds, and pipe transportation systems. The wide applications of terminal velocity have triggered active stud-ies in the past; various theoretical and experimental attempts have been made to investigate the sedimentation behavior of particles. Available results for the terminal velocity and the drag for the free setting of particles in Newtonian fluids are ample in the literature. In contrast, those in non-Newtonian fluids are relatively limited.

Fluids that exhibit non-Newtonian behavior are not un-common in practice. Blood, polymer solution, and emulsion are typical examples. Many empirical expressions have been

*_{Corresponding author. Fax: +886-2-23623040.} E-mail address:jphsu@ntu.edu.tw(J.-P. Hsu).

proposed for the description of these fluids[1]. For sedimen-tation in a non-Newtonian fluid, almost all of the available results are limited to spherical particles[2–5] and that for nonspherical particles are relatively limited[6–11]. Several attempts were made concerning the sedimentation in a vis-coelastic fluid, and behaviors that are different from those in a Newtonian fluid were observed (e.g.,[12–16]). The sedi-mentation of particles in a shear-thinning fluid has also been studied by many investigators. Under the creeping flow con-dition, Rodrigue et al.[16]derived the drag coefficients of both rigid spheres and bubbles in an infinite Carreau fluid. It was shown that the influence of the shear-thinning na-ture of a Carreau fluid on a rigid sphere is greater than that on a bubble. Turian [17] and Uhlherr et al. [18] investi-gated experimentally the sedimentation of a rigid sphere in a cylindrical tube filled with an inelastic power-law fluid. Navez and Walters[19]analyzed the settling of a sphere in shear-thinning polymer solutions. They pointed out that the influence of the shear-thinning nature of a fluid on the drag coefficient is greater than that of the elastic nature of the fluid. In a study of the wall effect on the motion of a particle

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under creeping flow condition Missirlis et al.[20]evaluated the drag coefficient for the case when a rigid sphere is in a cylinder filled with a power-law fluid. Ceylan et al.[21] pro-posed a theoretical model for the estimation of the drag force on a rigid sphere in a power-law fluid. The applicability of the model proposed was justified by experimental observa-tions[22], and it was found that it is applicable for Reynolds number up to 1000. Machaˇc et al.[23]analyzed the terminal falling velocity of spherical particles in a Carreau fluid under creeping flow conditions. Blackery and Mitsoulis[24] sim-ulated the sedimentation of a rigid sphere in a cylinder filled with Bingham plastic fluid under creeping flow condition. Pazwash and Robertson[25]measured the force acting on a disk in Bingham fluids. Torrest[26]claimed that Stokes’s law is capable of describing the settling of gravel chips in viscous non-Newtonian hydroxyethyl cellulose polymer so-lutions. Peden and Luo[27]reported the settling velocity of cylindrical and disk-shaped particles in drilling and fractur-ing fluids. Reynolds and Jones[28]measured the drag forces on various types of particles in non-Newtonian fluids. They concluded that using a volume-equivalent diameter is most appropriate in correlating their experimental data. Chhabra

[29]examined the settling of a cylindrical particle in a cylin-der filled with power-law liquids. Cho et al.[30]studied the sedimentation of a thin cylinder in both Newtonian and non-Newtonian fluids. General discussions on the subject are also available[31–33].

In this study, the sedimentation of a rigid cylindrical par-ticle in an infinite Carreau fluid is investigated for the case of a low to medium-large Reynolds number. In this case, since the governing equations for the flow filed are non-linear, solving them analytically becomes nontrivial. Here, a finite element scheme is adopted to circumvent the diffi-culty encountered. The influences of the key parameters of the problem under consideration, including the aspect ratio of a particle, the magnitude of the Reynolds number, and the nature of the fluid, on the drag coefficient are discussed.

2. Theory

Referring toFig. 1, we consider the sedimentation of an isolated, rigid cylindrical particle of diameter d and height h in a Carreau fluid. For convenience, the particle is held fixed and the surrounding fluid moves with bulk velocity ut, the terminal velocity of the particle. The cylindrical coordinates are adopted with their origin located at the center of the par-ticle; r and z are the radial and axial coordinates. At steady state, the flow field can be described by

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ρu· ∇u = −∇P + ∇ · τ,

(2) ∇ · u = 0,

where ρ is the density of fluid, P is the pressure,∇ is the gradient operator, τ is the stress tensor, and u is the fluid velocity. The constitutive equation for a generalized

New-Fig. 1. Sedimentation of a rigid, cylindrical particle of diameter d and height

h in an infinite Carreau fluid; ut is the terminal velocity of the particle. In mathematical modeling, the particle is held fixed and the fluid moves with bulk velocity ut. r and z are the radial and the axial coordinates.

tonian fluid can be expressed as

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τ= −η( ˙γ) ˙γ = −η( ˙γ)∇u + (∇u)T,

where ˙γ and ˙γ = [( ˙γ : ˙γ)/2]1/2are respectively the rate of strain tensor and its strength, η is the apparent viscosity, and the superscript T represents matrix transpose. Carreau pro-posed using[1]

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η(˙γ) = η_∞+ (η0− η∞)

1+ (λ ˙γ)2(n−1)/2,

where η0 and η∞ are respectively the apparent viscosities

corresponding to the minimum and the maximum ˙γ , λ is the relaxation time constant of the Carreau fluid, and n is its index parameter. Under conditions of practical significance, η0 η∞, and therefore, Eq.(4)reduces to

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η(˙γ) = η0

1+ (λ ˙γ)2(n−1)/2.

Note that if n→ 1 and/or λ → 0, the corresponding fluid is Newtonian.

The boundary conditions associated with Eqs.(1) and (2)

are assumed to be

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uz= ut as r→ ∞ or z → ∞,

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uz= 0 on particle surface,

where uzis the z-component of fluid velocity. The symmet-ric nature of the present problem also requires that

(8) ∂u

∂r = 0, r = 0.

For the present case, the drag force acting on a particle, FD, can be expressed as (9) FD= CD π d2 4 ρu2_z 2 ,

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where CD is the drag coefficient. For a general particle in

a Carreau fluid, CD is related to the shape and the

orien-tation of the particle and the nature of the fluid. The exact analytical expression for CDhas not been reported. Usually,

an empirical relation is assumed with its adjustable parame-ters estimated from experimental observations[34]. In the present study, the governing equations, Eqs.(1) and (2), and the associated boundary conditions, Eqs.(6)–(8), are solved numerically by FIDAP, a finite element scheme based on the Galerkin method. The drag force FD was calculated by

FI-DAP, and the drag coefficient CDis obtained by substituting

the calculated FDinto Eq.(9).

3. Results and discussion

The applicability of the present approach is examined by comparing the result obtained with that predicted by the semiempirical expression proposed by Machaˇc et al. [34], which is limited to low Reynolds numbers (Re0< 0.1). Fig. 2 shows the typical variation of CD as a function of

Re0. As can be seen from this figure, if Re0< 0.1, the

re-sult based on the present approach agrees well with that predicted by Machaˇc et al.[34]. On the other hand, if Re0

exceeds about 0.1, the deviation of the latter from the for-mer becomes appreciable. The deviation is positive and in-creases with Re0. This is expected because the inertial effect,

which is significant when Re0becomes large, is neglected

in Machaˇc et al. [34]. In general, the performance of the present approach is satisfactory. In subsequent discussions, the influences of the key parameters of the problem under consideration on the drag coefficient CDand on the flow field

are investigated through numerical simulation.

Fig. 3illustrates the variation of CDas a function of Re0

and Cu. This figure reveals that, in general, CD decreases

with the increase in both Cu and Re0. The former arises

from the fact that the larger the Cu the more important the shear-thinning nature of a fluid is, and the latter is similar to that for the case of a Newtonian fluid. If Re0 is

suffi-ciently small, log(CD) varies linearly with log(Re0) for both

the present Carreau fluid and the corresponding Newtonian fluid. The latter is expected because Stokes’s law is applica-ble if Re0is small. For both the present Carreau fluid and the

corresponding Newtonian fluid, a positive deviation from the Stokes’s law relation is observed if Re0is sufficiently large.

It is found that the smaller the Re0the more appreciable the

decrease of CDas Cu increases. For example, if n= 0.6, the

decreases in CDas Cu varies from 0.1 to 100 are respectively

70.81, 70.72, and 65.56% for Re0equals 0.1, 1, and 10.

Fur-ther calculations reveal that the influence of Cu on CD is

more significant at a smaller n than that at a larger n, which is expected since the smaller the n the more significant the shear thinning nature of a Carreau fluid is. For example, if

Re0= 0.1, the decreases in CDas Cu varies from 0.1 to 100

are 84.16, 70.81, and 42.36% for n equal to 0.4, 0.6, and 0.8, respectively.

Fig. 2. Variation of CDas a function of Re0for the case of a short cylinder in a Carreau fluid for the case when b= h/d = 0.5, Cu = 0.1, and n = 0.6. Solid line, semiempirical result of Machaˇc et al.[34]; discrete symbols, numerical result.

Fig. 3. Variation of CDas a function of Cu and Re0at various Cu for the case when b= h/d = 0.5 and n = 0.6.

Fig. 4illustrates the simulated variations of the drag co-efficient CD as a function of n at various combinations of

Cu and Re0. In general, CDincreases with the increases in

n, which is expected because the larger the n the less signif-icant the shear-thinning effect of a fluid is.Fig. 4indicates that if Cu is small, regardless of the value of Re0, the

influ-ence of n on CD is inappreciable. For example, if Cu= 1,

the increases in CDas n increases from 0.2 to 1.0 are 2.16,

4.20, and 11.46% for Re0= 0.1, 1, and 10, respectively. On

the other hand, if Cu is large, CDvaries appreciably as n

in-creases, even if Re0is small. For instance, if Cu= 10, the

increases in CD as n increases from 0.2 to 1.0 are

respec-tively 176.93, 178.20, and 213.03% for Re0= 0.1, 1, and 10.

These imply that, the influence of n on CD depends largely

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Fig. 4. Variation of CDas a function of n at various combinations of Cu and Re0for the case when b= h/d = 0.5 and n = 0.6. Curves 1–3, Cu = 1; 4–6, Cu= 10. Curves 1 and 4, Re0= 0.1; 2 and 5, Re0= 1; 3 and 6, Re0= 10.

Fig. 5. Variation of CDas a function of Cu and Re0at various b (= h/d) for the case when n= 0.6 and Cu = 0.1. Curve 1, b = 2; 2, b = 1; 3, b = 0.5.

Re0. Only if Cu is sufficiently large is the influence of n on

CDimportant.Fig. 4also indicates that for the same Re0, the

CD for a fluid with a large Cu approaches that with a small

Cu as n approaches unity, and it is possible that the former exceeds the latter. This is because the shear-thinning nature of the present Carreau fluid is governed by both λ and n.

Fig. 5shows the variation of the drag coefficient CDas a

function of Cu and Reynolds number Re0at various particle

aspect ratios b (= h/d). That as a function of n at various b is present inFig. 6. InFig. 5, because the cross-sectional area (or d) of a particle is fixed, the larger the b, the larger its lateral surface, the greater the drag exerted on the particle,

Fig. 6. Variation of CDas a function of n at various b (= h/d) for the case when n= 0.6, Cu = 10, and Re0= 10.

and therefore, a larger CD.Fig. 5suggests that log(CD)

var-ious nonlinearly with log(Cu). For a fixed b, CD decreases

with the increase in Cu, and for a fixed Cu, CD decreases

with the increase in b. The former arises from the fact that the larger Cu the more significant the shear-thinning effect is, and the latter is consistent with the previous discussion about the effect of b. The general trend observed inFig. 6

can be explained by similar reasoning.

The influences of Re0, Cu, and n on the flow field are

il-lustrated inFigs. 7–12.Figs. 7a and 7bindicate that if Re0

is small (<0.1), the flow field in the upstream of a particle is symmetric to that of its downstream. It becomes asym-metric, however, if Re0 is large (>0.1), as can be seen in Figs. 7c–7f. Here, because Re0 is not very large and the

liquid phase is infinitely large, the deformation of the flow field due to the presence of the particle is inappreciable. It is clearer by examining the variation of the vorticity shown inFig. 8, where the contours become asymmetric if Re0

ex-ceeds about 0.1. The flow field also becomes asymmetric if Cu is large, as suggested by the variation in fluid velocity illustrated in Fig. 9. This behavior arises from the shear-thinning nature of the Carreau fluid considered, since the larger the Cu the less viscous the fluid is, and it is easier for the fluid to separate from the particle surface in the down-stream.Figs. 10–12reveal that the influence of n on the flow field is not as appreciable as that of Cu and that of Re0. In

general, the influence of n, Cu, and Re0 on the flow field

follows the order n < Cu < Re0.

4. Conclusion

In summary, the sedimentation of an isolated cylindri-cal particle in a Carreau fluid is analyzed by evaluating the

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Fig. 7. Streamline contours at various Re0for the case when b= h/d = 0.5 and Cu = 1. (a) Re0= 0.05, (b) Re0= 0.1, (c) Re0= 0.5, (d) Re0= 1, (e) Re0= 5, (f) Re0= 10.

Fig. 8. Vorticity contours at various Re0for the case ofFig. 7.

drag coefficient under various conditions. We show that, in general, the qualitative behavior of the variation of the drag coefficient as a function of Reynolds number is similar to

that of a Newtonian fluid, that is, if Reynolds number is sufficiently small, a Stokes’s-law-like relation between drag coefficient and Reynolds number exists, and a positive

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de-Fig. 9. Velocity contours at various Cu for the case when b= h/d = 0.5 and Re0= 1. (a) Cu = 0.1, (b) Cu = 1, (c) Cu = 10, (d) Cu = 100.

Fig. 10. Streamline contours at different combinations of Re0and Cu for the case when b= h/d = 0.5. (a)–(d) Re0= 1 and Cu = 1; (e)–(h) Re0= 10 and Cu= 10. (a) n = 0.2, (b) n = 0.4, (c) n = 0.6, (d) n = 0.8, (e) n = 0.2, (f) n = 0.4, (g) n = 0.6, (h) n = 0.8.

viation from that law is observed when Reynolds number becomes large. If the relaxation time constant of the Carreau fluid is small, regardless of the value of Reynolds number, the influence of the index parameter of the Carreau fluid on the drag coefficient is inappreciable. On the other hand, if the relaxation time constant is large, CD varies

apprecia-bly as n increases, even if the Reynolds number is small. These imply that the influence of the index parameter on the drag coefficient depends largely on the magnitude of the re-laxation time constant, and is insensitive to the variation in Reynolds number. For the same Reynolds number, the drag coefficient for a fluid with a large relaxation time constant

approaches that with a small relaxation time constant as the index parameter approaches unity, that is, Newtonian fluid, and it is possible that the former exceeds the latter. As in the case of Newtonian fluids, if the Reynolds number is small, the flow field in the upstream of a particle is symmetric to that of its downstream. It becomes asymmetric, however, if the Reynolds number is large, where boundary separation may occur. The flow field also becomes asymmetric if the relaxation time constant becomes large. The influence of the index parameter on the flow field is not as appreciable as that of the relaxation time constant and that of the Reynolds number.

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Fig. 11. Vorticity contours at different combinations of Re0and Cu for the case ofFig. 10.

Fig. 12. Velocity contours at different combinations of Re0and Cu for the case when b= h/d = 0.5. (a)–(d) Re0= 1 and Cu = 10; (e)–(h) Re0= 10 and Cu= 1. (a) n = 0.2, (b) n = 0.4, (c) n = 0.6, (d) n = 0.8, (e) n = 0.2, (f) n = 0.4, (g) n = 0.6, (h) n = 0.8.

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Acknowledgment

This work is supported by the National Science Council of the Republic of China.

References

[1] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, sec-ond ed., Wiley, New York, 2002.

[2] R.P. Chhabra, C. Tiu, P.H.T. Uhlherr, Can. J. Chem. Eng. 59 (1981) 771–775.

[3] R.P. Chhabra, P.H.T. Uhlherr, Can. J. Chem. Eng. 58 (1980) 124–128. [4] B. Mena, O. Manero, L.G. Leal, J. Non-Newtonian Fluid Mech. 26

(1987) 247–275.

[5] C.K. Vassilios, Chem. Eng. Sci. 59 (2004) 4437–4447.

[6] R.P. Chhabra, K. Rami, P.H.T. Uhlherr, Chem. Eng. Sci. 56 (2001) 2221–2227.

[7] D. Rodrigue, R.P. Chhabra, D. DeKee, Can. J. Chem. Eng. 72 (1994) 588–593.

[8] Q. Xu, E.E. Michaelides, Int. J. Numer. Methods Fluids 22 (1996) 1075–1087.

[9] A. Tripathi, R.P. Chhabra, T. Sundararajan, Ind. Eng. Chem. Res. 33 (1994) 403–410.

[10] A. Tripathi, R.P. Chhabra, Am. Inst. Chem. Eng. J. 41 (1995) 728–731. [11] Y.J. Liu, D.D. Joseph, J. Fluid Mech. 255 (1993) 255–595.

[12] M.B. Bush, J. Non-Newtonian Fluid Mech. 55 (1994) 229–247. [13] J. Feng, D.D. Joseph, R. Glowinski, T.W. Pan, J. Fluid Mech. 283

(1995) 1–16.

[14] P.Y. Huang, J. Feng, D.D. Joseph, J. Fluid Mech. 271 (1994) 1–16. [15] P.Y. Huang, H.H. Hu, D.D. Joseph, J. Fluid Mech. 362 (1998) 297–

325.

[16] D. Rodrigue, D. DeKee, C.F. Chan Man Fong, Chem. Eng. Com-mun. 154 (1996) 203–215.

[17] R.M. Turian, Am. Inst. Chem. Eng. J. 13 (1967) 999–1006.

[18] P.H.T. Uhlherr, T.N. Le, C. Tiu, Can. J. Chem. Eng. 54 (1976) 497– 502.

[19] V. Navez, K. Walters, J. Non-Newtonian Fluid Mech. 67 (1996) 325– 334.

[20] K.A. Missirlis, D. Assimacopoulos, E. Mitsoulis, R.P. Chhabra, J. Non-Newtonian Fluid Mech. 96 (2001) 459–471.

[21] K. Ceylan, S. Herdem, T. Abbasov, Powder Technol. 103 (1999) 286– 291.

[22] A.M. Lali, A.S. Khare, J.B. Joshi, Powder Technol. 57 (1989) 39– 50.

[23] I. Machaˇc, B. Šiška, L. Machaˇccvá, Chem. Eng. Proc. 39 (2000) 365–369.

[24] J. Blackery, E. Mitsoulis, J. Non-Newtonian Fluid Mech. 70 (1997) 59–77.

[25] H. Pazwash, J.M. Robertson, J. Hydraul. Res. 13 (1975) 35–55. [26] R.S. Torrest, Am. Inst. Chem. Eng. J. 29 (1983) 506–508. [27] J.M. Peden, Y. Luo, SPE Drilling Eng. 2 (1987) 337–343.

[28] P.A. Reynolds, T.E.R. Jones, Int. J. Miner. Process. 25 (1989) 47– 77.

[29] R.P. Chhabra, Can. J. Chem. Eng. 70 (1992) 385–386.

[30] K. Cho, Y.I. Cho, N.A. Park, J. Non-Newtonian Fluid Mech. 45 (1992) 105–145.

[31] R. Clift, J. Grace, M.E. Weber, Bubbles, Drops and Particles, Academic Press, New York, 1978.

[32] R.P. Chhabra, Bubbles, Drops, and Particles in Non-Newtonian Fluids, CRC Press, Boca Raton, FL, 1993.

[33] G. McKinley, Transport Processes in Bubbles, Drops and Particles, second ed., Taylor & Francis, New York, 2002.

[34] I. Machaˇc, B. Šiška, R. Teichman, Chem. Eng. Proc. 41 (2002) 577–584.