Drag on a sphere in a spherical dispersion containing Carreau
fluid
J.P. Hsu
a,⁎
, S.J. Yeh
b,1, S. Tseng
ca
Department of Chemical Engineering and Institute of Polymer Science and Engineering, National Taiwan University, Taipei, Taiwan 10617 b
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 cDepartment of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137
A B S T R A C T
A R T I C L E I N F O Article history:
Received 29 November 2007
Received in revised form 17 March 2008 Accepted 24 March 2008
Available online 9 April 2008 Keywords:
Drag coefficient Spherical dispersion Unit cell model Carreaufluid
The drag on a rigid sphere in a spherical dispersion containing Carreaufluid is investigated theoretically based on a free surface cell model for Reynolds number in the range [0.1,100], Carreau number in the range [0,10], the power-law index in the range [0.3,1], and the void fraction in the range [0.271,0.999]. The influences of the particle concentration, the nature of the Carreau fluid, and Reynolds number, on the drag coefficient are examined. We show that the drag coefficient declines with the decreasing particle concentration, and the reversal of theflow field in the rear region of a sphere is enhanced by the shear-thinning nature of thefluid. An empirical relation, which correlates the drag coefficient with the void fraction (=1−particle concentration), the nature of the Carreau fluid, and Reynolds number, is proposed.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
The evaluation of the drag acting on a particle as it translates in a fluid medium is of both fundamental and practical significance. Sedimentation, which is often adopted to characterize the 2physical properties of a particle, for example, involves this type of problem. In a dispersion of particles the drag acting on an individual particle depends on its size, shape, relative density, and the concentration of particles. In particular, the influence of the presence of neighboring particles on the drag acting on a particle is of profound nature because it involves a many-body problem. In settling, for instance, the settling speed of a dispersion of particles is slower than that of an isolated particle[1]. Unfortunately, because particles seldom present indivi-dually in practice, that influence often needs to be considered and solving the associate problem becomes non-trivial. This difficulty can be circumvented by applying a cell model, in which a dispersion is simulated by a unit cell comprising a representative particle and a concentric liquid shell. Two types of cell model are available in the literature: free surface cell model[2]and zero vorticity cell model[3]. These models are essentially the same in all aspects except that zero shear stress and zero vorticity are assumed on the cell boundary in the former and in the latter, respectively.
The cell model has been applied successfully to simulate the behaviors and to calculate the physical properties of various types of dispersed system. These include, for instance, theflow through a
porous medium [4], permeability and viscosity studies [4], the pressure drop in packed and fluidized beds[4,5], the motion and mass transfer of a bubble swarm[6], the settling velocity of a rigid spherical dispersion[7], the sedimentation of rigid spheres[8], the mass transfer of spherical particles in assemblages [9–11], the sedimentation of a dispersion of spherical composite particles and thefluid flow through a bed of spherical composite particles[12], and the drag on a non-uniformly structuredfloc in a floc dispersion[13]. If the concentration of a dispersion is appreciable, it is usually of non-Newtonian nature[14–17]. In this case, because the viscosity of the dispersion is position dependent, the evaluation of the drag acting on a particle is much more complicated than that in the corresponding Newtonian case. Several attempts have been made to solve this type of problem. Kawase and Ulbrecht[18], for example, applied a free surface cell model and a boundary layer theory to analyze the behavior of an assemblage of rigid spheres in a power-lawfluids at high Reynolds numbers. Adopting free surface cell model [2] and variational principles, Chhabra and Raman[19]were able to obtain bounds on the drag for the creepingflow of a Carreau fluid past an assemblage of rigid spheres. Staish and Zhu[20] and Jaiswal et al.[21,22]solved numerically the problem of an unbounded slow flow of non-Newtonianfluids (power-law or Carreau model) through an assem-blage of rigid spheres. Using both a free surface cell model and a zero vorticity cell model, Ferreira et al.[23,24]considered the steadyflow of an incompressible power-lawfluid across an assemblage of rigid cylinders. Note that both the free surface and zero vorticity cell models are idealizations, and there is no justification for either of these boundary conditions. In a discussion of cell models, Zholkovskiy et al.
[25]pointed out that if the particle volume fraction is sufficiently low, the zero vorticity cell model performs better than the free surface cell
⁎ Corresponding author. Fax: +886 2 23623040.
E-mail addresses:jphsu@ntu.edu.tw(J.P. Hsu),f92524027@ntu.edu.tw(S.J. Yeh),
topology@mail.tku.edu.tw(S. Tseng). 1 Fax: +886 2 23623040.
0032-5910/$– see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2008.03.014
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Powder Technology
model does. However, if the volume fraction approaches the value attributed to the close packing, the free surface cell model becomes the better choice among other models, including the zero vorticity cell model.
In this study, the drag on a rigid sphere in a spherical dispersion containing Carreaufluid is estimated based on a free surface cell model for the case of small to medium large Reynolds number. The influences of the concentration of particles, the Reynolds number, and the properties of a Carreau fluid on the drag coefficient are in-vestigated. Empirical relationships that correlate the drag coefficient with the key parameters of the system under consideration are proposed.
2. Theory
Let us consider the steady translation of rigid spheres of radius rp
in a liquid. For convenience, the spheres arefixed in the space and the velocity of the approaching liquid is V. Referring toFig. 1, a unit cell model is adopted where the spherical dispersion is simulated by a cell comprising a representative sphere and a concentric spherical liquid shell of radius Rc. The cylindrical coordinates are adopted with its
origin located at the center of the cell, and r and z are the radial and the axial coordinates, respectively. The concentration of the spherical dispersion can be estimated by (1−ε), where ε is the void fraction defined by
e ¼ 1 rp
Rc
3
: ð1Þ
Suppose that the liquid phase is incompressible. Then theflow field can be described by
qu ju ¼ jP þ j t ð2Þ
j u ¼ 0; ð3Þ
whereρ is the density of the liquid phase, P is the pressure, ▽ is the gradient operator,τ is the stress tensor, and u is the liquid velocity. For a generalized Newtonianfluid[26–29]
t ¼ g :gð Þ :γ ð4Þ
whereγ· =(▽u+(▽u)T) is the rate-of-strain tensor,γ˙ is its magnitude,
η is the apparent viscosity, and the superscript T denotes matrix transpose. Suppose that the liquid phase is a Carreau fluid, the viscosity of which can be described by[26–29]
g :gð Þ ¼ g0 1þ k :gð Þ2
h iðn1Þ=2
ð5Þ whereη0is the viscosity corresponding to the minimum shear rate, k
is the relaxation time constant, and n is the power-law index. Note that if n = 1 and/or k = 0, the liquid becomes Newtonian.
The following boundary conditions are assumed for theflow field:
uz¼ V at r ¼ Rc ð6Þ
srh¼ 0 at r ¼ Rc ð7Þ
uz¼ 0 on the sphere surface ð8Þ
where uzis thefluid velocity in the z-direction.
3. Results and discussion
FIDAP7.6, a commercial software based on afinite element method, is adopted to solve the governing equations and the associated boundary conditions. Throughout the computation, double precision is used and grid independence is checked. In our case, using roughly 7000 elements in the liquid domain is sufficient.Fig. 2illustrates the typical mesh used. The applicability of the software adopted is justified by applying it to the case of a Newtonianfluid and comparing the result obtained with the available result in the literature. The drag acting on an isolated rigid sphere in afluid, FD, can be expressed as[26,30–32]
FD¼ 1 2qV 2 pr2 p CD ð9Þ
where CDis the drag coefficient.Fig. 3shows the variation of FDas a
function ofε for the case of a Newtonian fluid under a creeping flow condition. Both the analytical result of Happel and Brenner[4]and the numerical result based on the present approach are presented. As seen inFig. 3, the performance of the software adopted is satisfactory.
For illustration, we assume thatη0= 3 g/cm s, and therefore the
parameters key to the behavior of a sphere are the Reynolds number, the concentration of spheres, and the properties of the Carreaufluid. The influences of these parameters on the flow field and the drag coefficient are investigated through numerical simulation. For convenience, we define the Carreau number Cu and the Reynolds number Re as[26–29]
Cu ¼ kV=rp ð10Þ
Re ¼ 2qrpV=g0: ð11Þ
The simulatedflow fields near a sphere at various combinations of the void fractionε, the Carreau number Cu, and the Reynolds number
Fig. 1. Steady translation of rigid spherical particles of radius rpin a Carreaufluid. For convenience, the spheres arefixed in the space and the velocity of the approaching liquid is V. The system is simulated by a unit cell model where it is mimicked by a cell comprising a representative sphere and a concentric spherical liquid shell of radius Rc. The cylindrical coordinates are adopted with its origin located at the center of the cell, and r and z are the radial and the axial coordinates, respectively.
Re are illustrated in Figs. 4–6. These figures suggest that if Re is sufficiently small, the flow field in the front region of a sphere is essentially symmetric to that in its rear region, regardless of the
levels of ε and Cu. However, as Re becomes large, they are no longer symmetric to each other. In this case, a reverse flow is present and vortexes can be observed in the rear region of the sphere. The presence of the neighboring spheres has the effect of confining this reverse flow so that the reverse of the flow field in the rear region of a sphere occurs only ifε is sufficiently large. On the other hand, the shear-thinning nature of thefluid has the effect of enhancing the reverse flow. Note that if the shear-thinning nature of thefluid is significant, such as inFig. 6(d), althoughε is not large enough to observe the reverse flow in the rear region of the sphere, the flow field there becomes unstable, and local turbulentflow is present.
Fig. 7 shows the influence of particle concentration on the drag coefficient CD for various combinations of Cu and Re. This
figure reveals that for the range of Re considered, regardless of the magnitude of Re, the qualitative behaviors of CD are roughly
the same. In general, for a fixed Cu, CD decreases with the
in-creasing in ε (decreasing in the particle concentration), and the smaller the ε (higher the particle concentration) the more sig-nificant is its influence on CD. For afixed ε, CDdeclines with the
increase in Cu, which is expected because the larger the Cu the more important the effect of shear thinning is. Also, the larger theε (lower the particle concentration) the more significant the
Fig. 4. Flowfields for various combinations of ε and Cu at n=0.6 and Re=0.1. Cu=0.1 in (a), (b), and (c), and Cu=10 in (d), (e), and (f). (a) and (d), ε=0.271, (b) and (e), ε=0.657, (c) and (f),ε=0.999.
Fig. 3. Variation of the drag on a rigid sphere in a spherical dispersion of Newtonian fluid as a function of ε for various values of (rp/ Rc) at Re = 0.01. Solid curve, result of Happel and Brenner[4], discrete symbols, present numerical results.
influence of Cu is, and this phenomenon is pronounced if n be-comes smaller.
The influence of the particle concentration on the drag coef-ficient, measured by the ratio R= [CD(ε= 0.271)/CD(ε= 0.999)], at
various combinations of n, Re, and Cu is summarized inTable 1. As seen from this table, the influence of the particle concentration is more important at a smaller Cu, a smaller Re, and a larger n, implying that the shear-thinning nature of afluid has the effect of reducing the influence of particle concentration. It is interesting to observe that at n = 0.6 and Cu = 10, R has a local minimum as Re varies. A local minimum in R is also observed as Cu varies at n = 0.6 and Re = 100, and as Re varies at both Cu = 0.5 and Cu = 1 for n = 0.5, 0.4, and 0.3. In general, the more important the shear-thinning nature of a fluid the easier for R to have a local minimum. The presence of the local minimum arises from the specific behavior of theflow field in the rear region of a sphere such as that shown in
Fig. 6(d).
From application point of view, it is highly desirable to correlate CDwith the parameters key to the system under consideration. To
this end, a regression analysis is conducted based on the simulation
data gathered in our study. For 0.5bn≦0.8, 0.1≦Cu≦10, ε≦0.7, and Re≦100, we arrive at
CD¼
A
CuðBCReÞReDe½EþF expðRe=GÞ ð12Þ
A¼ 11:7 þ 170n ð12aÞ B¼ 0:9455 0:935n ð12bÞ C¼ 0:00617 0:007n ð12cÞ D¼ 0:844 þ 0:12n ð12dÞ E¼ 2:72384 2:06055 1012 exp n 0:02835 ð12eÞ F¼ 2:33773 107 exp n 0:04779 ð12fÞ G¼ 3:19373 105 exp n 0:0456 : ð12gÞ
Fig. 5. Flowfields for various combinations of ε and Cu at n=0.6 and Re=40. Cu=0.1 in (a), (b), and (c), and Cu=10 in (d), (e), and (f). (a) and (d), ε=0.271, (b) and (e), ε=0.657, (c) and (f),ε=0.999.
For 0.3≦n≦0.5, 0.1≦Cu≦1, ε≦0.7, and Re≦40, we obtain CD¼ AV CuðBVC VReÞReDVeEV ð13Þ AV¼ 43:333 þ 110n ð13aÞ BV¼ 0:90533 0:86n ð13bÞ CV¼ 0:00547 0:007n ð13cÞ DV¼ 0:89133 þ 0:06n ð13dÞ EV¼ 1:47333 þ 2:05n: ð13eÞ For 0.8bn≦0.9, 0.1≦Cu≦10, ε≦0.7, and Re≦100, CD can be
de-scribed by
CD¼
169
Cuð0:0990:0002ReÞRe0:954eð3:6560:0031ReÞ: ð14Þ Fig. 8summarizes the variation of the CDbased on the correlation
relationships, Eqs. (12)–(14), as a function of the CDcalculated in this
study. In general, the performance of the correlation relationships obtained is satisfactory. Although the maximum deviation is on the order of 30%, only few points deviate appreciably. In general, the larger the value of n the better the performance of Eqs. (12)–(14).
The performance of the present model is further examined by comparing the results calculated with those predicted by other theoretical models[20,21,33,34], where a power law, shear-thinning fluid in the creeping flow region was considered with
g :gð Þ ¼ Kj :gjm1: ð15Þ
The values of the parameters assumed are K = 20 g/(cm2 s) and
m = 0.6, which correspond approximately toη0= 100 poise,λ=55.90 s,
and n = 0.6 in the present Carreau model. For Ren=ρV2− m(2rp)m/ K≪1,
a drag correction factor X is defined as[35]
X eð; nÞ ¼CDRen
24 : ð16Þ
Fig. 9illustrates the variation of X as a function ofε based on the present model and that based on the other theoretical models. As seen, the general trend of X is consistent with that predicted by other
Fig. 6. Flowfields for various combinations of ε and Cu at n=0.6 and Re=100. Cu=0.1 in (a), (b), and (c), and Cu=10 in (d), (e), and (f). (a) and (d), ε=0.271, (b) and (e), ε=0.657, (c) and (f),ε=0.999.
models.Fig. 9reveals that the larger theε the closer the X evaluated by the present model to that of the other models. This might arise from that the higher the concentration of particles (ε is smaller) the greater the difference between the nature of the present Carreaufluid and that of the corresponding power-lawfluid. This can be justified by the experimental observation of Machač and Lecjaks[36], where the terminal velocity of a sphere in a rectangular ductfilled with a non-Newtonianfluid is studied. Kerafloc, the polymer solution used in theirFig. 5, can be described by a Carreaufluid. They found that if the boundary effect is insignificant, the wall factor for the case of a power-lawfluid[37]is close to that of a Carreaufluid. On the other hand, if the boundary effect is significant, then the difference between the two is appreciable. These imply that the smaller the ε the greater the difference between the X calculated by the present model and that predicted by the other power-law based models, as is seen inFig. 9.
4. Conclusion
The drag on a rigid sphere in a spherical dispersion containing Carreaufluid is estimated based on a free surface cell model. Nu-merical simulations are conducted for Reynolds number in the range [0.1,100], Carreau number in the range [0,10], the power-law index in the range [0.3,1], and the void fraction in the range [0.271,0.999]. We show that if Reynolds number is sufficiently small, the flow field in the front region of a sphere is essentially symmetric to that in its rear region; if it is large, a reverseflow is present in the rear region of the sphere and theflow fields are no longer symmetric. The presence of the neighboring spheres has the effect of confining this reverse flow, but the shear-thinning nature afluid has the effect of enhancing that flow. The drag coefficient declines with the decreasing in the particle concentration and the higher the particle concentration the more
significant is its influence on the drag coefficient. An increase in the Carreau number has the effect of reducing the drag coefficient, and the lower the particle concentration and/or the smaller the power-law index the more significant that effect is. The degree of influence of the particle concentration on the drag coefficient may have a local minimum as Carreau number or Reynolds number varies; the more important the shear-thinning nature of a fluid the easier for that influence to have a local minimum. Empirical relationships are de-veloped, which correlate the drag coefficient with Reynolds number, Carreau number, the power-law index, and the particle concentration.
Acknowledgment
This work is supported by the National Science Council of the Republic of China.
References
[1] B.H. Kaye, R.P. Boardman, Proceeding Symposium Interaction between Fluids and Particles, British Institute of Chemical Engineers, London, 1962.
[2] J. Happel, Viscousflow in multiparticle systems-slow motion of fluids relative to beds of spherical particles, AIChE Journal 4 (1958) 197–201.
[3] S. Kuwabara, The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscousflow at small Reynolds numbers, Journal of the Physical Society of Japan 14 (1959) 527–532.
[4] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Academic Press, New York, 1983.
[5] M.M. Elkaissy, G.M. Homsy, Theoretical study of pressure-drop and transport in packed-beds at intermediate Reynolds-numbers, Industrial & Engineering Chem-istry Fundamentals 12 (1973) 82–90.
[6] B.P. LeClair, A.E. Hamielec, Viscous flow through particle assemblages at intermediate Reynolds-numbers— cell model for transport in bubble swarms, Canadian Journal of Chemical Engineering 49 (1971) 713–720.
[7] G.K. Batchelor, Sedimentation in a dilute dispersion of spheres, Journal of Fluid Mechanics 52 (1972) 245–268.
[8] A.K. Jaiswal, T. Sundararajan, R.P. Chhabra, Hydrodynamics of Newtonian fluid-flow through assemblages of rigid spherical-particles in intermediate Reynolds-number regime, International Journal of Engineering Science 29 (1991) 693–708. [9] Z.S. Mao, Numerical simulation of viscous flow through spherical particle assemblage with the modified cell model, Chinese Journal of Chemical Engineering 10 (2002) 149–162.
[10] Z.S. Mao, J.Y. Chen, Numerical approach to the motion and external mass transfer of a drop swarm by the cell model, Proc. ISEC 2002, Capetown, South Africa, 2002. [11] Z.S. Mao, Y. Wang, Numerical simulation of mass transfer of a spherical particle
assemblage with the cell model, Powder Technology 134 (2003) 145–155. [12] H.J. Keh, Y.C. Chang, Creeping motion of an assemblage of composite spheres
relative to afluid, Colloid and Polymer Science 283 (2005) 627–635.
[13] J.P. Hsu, M.C. Li, A.C. Chang, Drag of a dispersion of nonhomogeneously structured flocs in a flow field, Journal of Colloid and Interface Science 284 (2005) 332–338. [14] R.P. Chhabra, C. Tiu, P.H.T. Uhlherr, Creeping motion of spheres through Ellis model
fluids, Rheologica Acta 20 (1981) 346–351.
[15] A. Tripathi, R.P. Chhabra, Slow power lawfluid-flow relative to an array of infinite cylinders, Industrial & Engineering Chemistry Research 31 (1992) 2754–2759. Table 1
Influence of the particle concentration on the drag coefficient, measured by the ratio R= [CD(ε=0.271)/CD(ε=0.999)], at various combinations of n, Re, and Cu
N Re Cu R 0.9 0.1 0.1 732.30 1 590.64 10 530.50 10 0.1 502.72 1 408.75 10 365.86 40 0.1 309.78 1 256.27 10 229.74 100 0.1 212.00 1 180.63 10 165.16 0.8 0.1 0.1 554.26 1 361.14 10 292.85 10 0.1 380.71 1 252.29 10 203.96 40 0.1 235.74 1 163.23 10 135.45 100 0.1 164.67 1 124.15 10 112.12 0.7 0.1 0.1 419.00 1 221.10 10 163.14 10 0.1 288.00 1 156.19 10 115.91 40 0.1 179.66 1 106.62 10 88.45 100 0.1 129.12 1 90.96 10 88.39 0.6 0.1 0.1 316.20 1 134.89 10 91.10 10 0.1 217.57 1 97.27 10 68.93 40 0.1 137.24 1 72.75 10 67.40 100 0.1 102.00 1 72.00 10 80.00 0.5 0.1 0.1 238.00 0.5 109.71 1 82.57 10 0.1 164.04 0.5 78.22 1 61.26 40 0.1 105.20 0.5 59.13 1 53.08 100 0.1 82.17 0.5 60.26 1 59.82 0.4 0.1 0.1 178.51 0.5 70.66 1 50.61 10 0.1 123.31 0.5 51.62 1 39.46 40 0.1 80.84 0.5 44.60 1 41.40 100 0.1 67.23 0.5 49.82 1 49.39 0.3 0.1 0.1 133.06 0.5 45.30 1 30.53 Table 1 (continued) N Re Cu R 0.3 10 0.1 92.26 0.5 34.30 1 26.39 40 0.1 62.19 0.5 34.50 1 33.09
Fig. 8. Variation of the CDbased on the correlation relationships, Eqs. (12)–(14), as a function of the CDcalculated in this study.
Fig. 9. Variation of the drag correction factor (X) of non-Newtonianfluid as a function of ε for the case where Ren= 0.01,η0= 100 poise, k = 55.90 s, and n = 0.6. Solid curve, result predicted by the present analysis; (■) Jaiswal et al.[21]; (●) Satish and Zhu[20]; (▲) Kawase and Ulbrecht[33]; (▼) Mohan and Raghuraman[34].
[16] M.V. Bruschke, S.G. Advani, Flow of generalized Newtonianfluids across a periodic array of cylinders, Journal of Rheology 37 (1993) 479–498.
[17] Y.J. Liu, D.D. Joseph, Sedimentation of particles in polymer-solutions, Journal of Fluid Mechanics 255 (1993) 565–595.
[18] Y. Kawase, J.J. Ulbrecht, Motion of and mass-transfer from an assemblage of solid spheres moving in a non-Newtonianfluid at high Reynolds-numbers, Chemical Engineering Communications 8 (1981) 233–249.
[19] R.P. Chhabra, J.R. Raman, Slow non-Newtonianflow past an assemblage of rigid spheres, Chemical Engineering Communications 27 (1984) 23–46.
[20] M.G. Satish, J. Zhu, Flow resistance and mass-transfer in slow non-Newtonianflow through multiparticle systems, Journal of Applied Mechanics 59 (1992) 431–437. [21] A.K. Jaiswal, T. Sundararajan, R.P. Chhabra, Hydrodynamics of creepingflow of power lawfluids through particle assemblages, International Journal of Engineer-ing Science 31 (1993) 293–306.
[22] A.K. Jaiswal, T. Sundararajan, R.P. Chhabra, Slow non-Newtonianflow through packed beds effect of zero shear viscosity, The Canadian Journal of Chemical Engineering 71 (1993) 646–651.
[23] J.M. Ferreira, R.P. Chhabra, Analytical study of drag and mass transfer in creeping power lawflow across tube banks, Industrial & Engineering Chemistry Research 43 (2004) 3439–3450.
[24] A.A. Soares, J.M. Ferreira, R.P. Chhabra, Steady two-dimensional non-Newtonian flow past an array of long circular cylinders up to Reynolds number 500: a numerical study, Canadian Journal of Chemical Engineering 83 (2005) 437–449. [25] E.K. Zholkovskiy, V.N. Shilov, J.H. Masliyah, M.P. Bondarenko, Hydrodynamic cell
model: general formulation and comparative analysis of different approaches, Canadian Journal of Chemical Engineering 85 (2007) 701–725.
[26] R.P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Taylor & Francis, Boca Raton, FL, 2007.
[27] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Wiley, New York, 1977.
[28] H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology, Elsevier, New York, 1989.
[29] J. Ferguson, Z. Kemblowski, Applied Fluid Rheology, Elsevier, New York, 1991. [30] R. Clift, J. Grace, M.E. Weber, Bubbles, Drops and Particles, Academic Press,
New York, 1978.
[31] R.M. Turian, An experimental investigation offlow of a aqueous non-Newtonian high polymer solutions past a sphere, AIChE Journal 13 (1967) 999–1006. [32] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley and Sons,
New York, 2002.
[33] Y. Kawase, J.J. Ulbrecht, Drag and mass transfer in non-Newtonianflows through multi-particle systems at low Reynolds numbers, Chemical Engineering Science 36 (1981) 1193–1202.
[34] V. Mohan, J. Raghuraman, Bounds on the drag for creepingflow of an Ellis fluid past an assemblage of spheres, International Journal of Multiphase Flow 2 (1976) 581–589.
[35] R.P. Chhabra, J. Comiti, I. Machač, Flow of non-Newtonian fluids in fixed and fluidised beds, Chemical Engineering Science 56 (2001) 1–27.
[36] I. Machač, Z. Lecjaks, Wall effect for a sphere falling through a non-Newtonian fluid in a rectangular duct, Chemical Engineering Science 50 (1995) 143–148. [37] P.V. Balaramakrishna, R.P. Chhabra, Sedimentation of a sphere along the axis of a
long square ductfilled with non-Newtonian liquids, The Canadian Journal of Chemical Engineering 70 (1992) 803–807.
