7664
Ind. Eng. Chem. Res. 2004, 43, 7664-7670
Experiment on Settling of Slender Particles with Large Aspect Ratio and Correlation of the Drag Coefficient Long Fan,†,‡ Zai-Sha Mao,*,† and Chao Yang† Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China, and Department of Environmental Science and Engineering, Tsinghua University, Beijing 100084, China
The drag coefficient of slender particles is an important parameter to operations involved with them. In this paper, settling of slender particles with a large aspect ratio and of different materials in laminar Newtonian fluids is studied. It is observed that the particle orientation oscillates during settling and stabilizes to its equilibrium horizontal orientation eventually. Experiments are conducted in different fluids (water and glycerin solutions, quiescent or flowing) for particles with different densities (from 1125 to 8000 kg‚m-3) and aspect ratios (from 4 to 50). It is found that the product of the drag coefficient and cosine of the orientation angle is independent of the particle aspect ratio and the bulk liquid flow. Instead of using the concept of sphericity, the drag coefficient of slender particles is effectively correlated with the orientation and the Reynolds number based on the particle cylindrical diameter. With the aid of the modified Archimedes number, the proposed general correlation CD cos φ ) (24/Re)(0.006983 + 0.6224Re-1.046)(Fp/Fl)-1.537(Ar*)0.8524 represents very well the experimental data under various experimental conditions with the average relative error of 9.4%. 1. Introduction Spherical particles are widely encountered in numerous cases in daily life and process industries. Considerable achievements have been obtained through theoretical analysis, extensive experiments,1 and computer simulations.2,3 In contrast to the vast literature available on spherical particles, much less work was reported on nonspherical particles. Among them, slender particles, especially fibrous slender particles, play important roles in many important industrial processes, such as papermaking and synthetic cellulose. In recent years, the problems related to slender particles attracted increasing attention from the process engineering community. For slender particles, Liu and Joseph4 studied the effects of liquid properties, particle density, length, and shape on the settling velocity and orientation. They found that both the particle concentration in liquid and the shape of the two ends of a particle influenced the particle orientation during settling. For cylinders of an aspect ratio from 0.25:1 to 5.0:1, Mckay et al.5 investigated the influence of the aspect ratio on the orientation, as well as the effects of the aspect ratio and viscosity of the fluid on the settling velocity. They concluded that when the aspect ratio was greater than 1, the terminal velocity would approach a constant as the aspect ratio increases. Turney et al.6 experimented with synthetic fiber (Fp ) 1950 kg‚m-3) and obtained the slurry sedimentation velocity as a function of φ, terminal settling velocity, and aspect ratio:
U ) U0f(φ,up,r)
0.035 < φ < 0.118
(1)
where φ is the initial suspension volume fraction. * To whom correspondence should be addressed. Tel.: +8610-62554558. Fax: +86-10-62561822. E-mail: zsmao@ home.ipe.ac.cn. † Chinese Academy of Sciences. ‡ Tsinghua University.
Owing to the prominent shape effect, the drag coefficient of slender particles is different from that of spherical particles. However, the literature contains no sufficient information on this issue. Currently available methodologies to deal with this problem fall into two distinct categories. In the first approach, effort is directed to the development of a single correlation for all nonspherical particles with different shapes and orientation. Most of these studies employed the so-called volume equivalent sphere diameter, d, as the characteristic size in combination with the sphericity to quantify the shape effect. Thus, correlations of this kind are in the form of
f(Re,CD,ψ) ) 0
(2)
where Re is the particle Reynolds number (Re ) d|ul up|Fl/µ), CD is the drag coefficient, and ψ is the sphericity of the particles. By this method, Haider and Levenspiel7 proposed the following general form of the drag coefficient for spherical and nonspherical particles in incompressible media:
CD )
24 C (1 + AReB) + Re 1 + D/Re
(3)
where the values of A, B, C, and D were obtained by fitting the calculated and experimental data on the drag coefficient. For nonspherical particles, the constants A, B, C, and D were found to be a function of sphericity. This method is very common in dealing with the drag coefficient for nonspherical particles for its advantage of simplicity. However, because the resulting correlation is usually based on the database to which spherical particles contribute most, it is less accurate for slender particles. The second approach involves the development of a drag expression for particles of fixed shape and orientation, such as cylinders, needles, disks, etc. The use of this approach is limited, and it seems less common than
10.1021/ie049479k CCC: $27.50 © 2004 American Chemical Society Published on Web 10/06/2004
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7665
the first. However, it appears more accurate for nonspherical particles. For nonspherical particles with special shape, as is for spherical particles, study of the drag coefficient with a fixed orientation of particles is relatively simple. Blumberg and Mohr8 worked on the drag coefficientReynolds number relationship for free falling of cylinders with their axis normal to the direction of motion. Kasper et al.9 reported experimental values of the drag coefficient for freely settling tungsten cylinders in viscous silicone oil with their particle axis in line with the direction of motion. Unnikrishnan and Chhabra10 studied the terminal velocity of cylinders with their axis parallel to the direction of motion in terms of the drag coefficient and Reynolds number. All of the studies mentioned above focused on a settling particle with a fixed orientation or terminal settling velocity without explicit consideration of the particle orientation as a variable. In general, the axis of particles is either normal or parallel to the direction of motion. Besides, the drag coefficient of slender particles with a large aspect ratio (r > 10) has not been thoroughly studied. The aim of this paper is to determine the drag coefficient of slender particles during acceleration and steady motion through stagnant and/or flowing liquid. The data show an obvious correlation with the particle orientation. A new correlation of the drag coefficient is established, and it agrees well with the present experimental data.
When a slender rigid particle is settling under gravity, it is controlled by the force balance:
dup d(ul - up) F ) Fpg - Flg + CvmFl dt Vp dt
S ) ld cos φ ) rd2 cos φ
(6)
The acceleration term in eq 4 is to be replaced by a quotient of differences in order to calculate CD. Thus,
[( )
]
Fl ∆up ∆(ul - up) F (7) ) Fp 1 g+ CvmFl Vp Fp ∆t ∆t The combination of eqs 5-7 yields
( ) (
)
Fp Fp ∆up -1 g+ Cvm Fl Fl ∆t
(5)
where CD is the drag coefficient and S is the projected area of the particle normal to the direction of settling and a function of the aspect ratio and orientation. Orientation φ is defined as the angle between the major axis of slender particles and the horizontal direction, as shown in Figure 1. A slender particle is never observed to keep moving in a Newtonian fluid with an orientation of φ ) 90° after the particle is released. Therefore, the drag coefficient for φ ) 90° is not considered here.
(8)
When the particle falls steadily, ∆up ) 0.
( )
Fp -1 g Fl
π CD ) d 2 |ul - up|(ul - up) cos φ
(4)
On the right side of eq 4, gravity, drag force, buoyancy, and virtual mass force are all present. Here, Vp is the particle volume, Vp ) πld2/4, and Cvm is the virtual mass coefficient, which is approximately taken as 0.5.11 The contribution of the virtual mass force should be included, especially while the motion of the particle is in the acceleration stage through a liquid medium. Cvm equal to 0.5 is tentatively adopted throughout this work because no general recommendation for nonspherical particles in laminar flow has been addressed in the literature. The definition of the drag force is
Fl F ) CDS |ul - up|(ul - up) 2
For φ * 90°
π CD ) d 2 |ul - up|(ul - up) cos φ
2. Calculation of the Drag Coefficient from Settling Experiment
Fp
Figure 1. Orientation of a slender particle.
(9)
In contrast, the expression for calculating the drag coefficient of spherical particles from measurement is
( ) (
4 CDS ) d 3
)
Fp Fp ∆up -1 g+ Cvm Fl Fl ∆t |ul - up|(ul - up)
( )
Fp -1 g Fl
4 CDS ) d 3 |ul - up|(ul - up)
(accelerating) (10)
(steady motion) (11)
3. Experimental Section The experimental system to measure the drag coefficient of slender particles is shown in Figure 2. The diameter of the settling column (6) is 187 mm, which is wide enough to avoid the wall effect on the sedimentation of slender particles. A slender particle is released beneath the liquid surface at the top of the column. The pump (10) is operated when the effect of liquid flow is examined, and valves (8 and 12) are then set open. The settling process is recorded by a CCD digital camera (LG CCD GC-145C-G), and the images are handled by an image sampling card (DH-VRT-CG200, made by Daheng
7666
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004
Figure 2. Diagram of the experimental apparatus: (1) zoom lens; (2) CCD camera; (3) image sampling card; (4) computer; (5) slender particle; (6) column; (7) light; (8) valve; (9) flowmeter; (10) peristaltic pump; (11) tank; (12) valve.
Co., Beijing, China) and then stored in a PC computer. Because the settling velocity of the particles varies, the shutter of the CCD is changed between 1/60 s for synthetic fiber and 10-5 s for brass wire to get sharp images of particles. Slender particles (brass wire, Fp ) 8000 kg‚m-3, d ) 0.155 mm; synthetic fiber A, Fp ) 1125 kg‚m-3, d ) 0.160 mm; synthetic fiber B, Fp ) 1246 kg‚m-3, d ) 0.370 mm) with different aspect ratios from 4 to 50 have been used in this study. In addition, glass spheres are also used to check the overall accuracy and reliability of the experimental method in this work. The diameters of the particles are measured using a micrometer reading up to 0.001 mm. The density of each particle is measured using a picnometer. The experiments are performed in water and glycerin (9.9 wt %) solutions (Fl ) 1055 kg‚m-3, µ ) 1.318 × 10-3 kg‚m-1‚s-1) at ambient conditions.
Figure 3. Drag coefficient of spheres in water.
4. Results of Spherical Particles The experimental procedure employed in this study is checked by a preliminary test on spherical particles. Experiments are carried out on the settlement of spherical particles (glass beads) in stagnant water. The diameter of glass beads is 2.25 mm with a density of 2585 kg‚m-3. The drag coefficient is calculated by eqs 10 and 11 using the data of particle position versus time obtained by reading the CCD images frame by frame. The drag coefficients are plotted against Re in Figure 3. Also shown for comparison are the curves of empirical correlations (12)1 and (13):12
CD ) CD )
24 (1 + 0.15Re0.687) Re
0 < Re < 1000
(12)
24 0.413 [1 + 0.173Re0.657] + Re 16300Re-1.09 0 < Re < 2 × 105 (13)
In Figure 3, the measurements of the drag coefficient calculated using eqs 10 and 11 are in good agreement with the literature correlations. It is noticed that the data consist partly of particles in acceleration. Therefore, it is necessary to take into account the virtual mass force. The value of the virtual mass coefficient taken to be 0.5 is acceptable. Verified by this comparison, it is believed that the method used in this paper gives a reliable drag coefficient for slender particles. 5. Results of Slender Particles 5.1. Orientation. Figure 4 shows the variation of a settling slender particle in stagnant water. It is found
Figure 4. Images of a settling synthetic fiber A (d ) 0.160 mm) in stagnant water.
that particles in stagnant liquid would turn around its center of gravity during settling. The orientation of a particle approaches gradually from whatever its original orientation is to a final stable horizontal orientation, as shown in Figure 5. Huang et al.13 reported the behavior of elliptical spheroids and found that they would oscillate around a stable orientation during settling in stagnant Newtonian fluid, which is consistent with our observations. The particles settling in flowing liquid also show the same results as those illustrated in Figure 6. From this figure, it can be concluded that the orientation is irrespective of the fluid velocity and the slender particle would oscillate and turn to a horizontal orientation eventually. This is easy to understand if the relative motion of particles with respect to liquid is used to interpret the phenomena. 5.2. Effect of Fluid Flow. Settling experiments may be conducted in three ways: with either ul > 0, ul < 0, or ul ) 0. The drag coefficients are calculated with eq 8. It is found that the drag coefficient (CD) is very poorly
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7667
Figure 5. Variation of orientation for a synthetic fiber A (d ) 0.160 mm) in stagnant water.
sharply with the increasing Reynolds number. Note that the Reynolds number is defined based on the relative velocity and slender particle cylinder diameter d. This definition makes the bulk liquid velocity irrelevant to the correlation of CD cos φ with Re. 5.3. Effect of the Aspect Ratio. Figures 8-10 illustrate the influence of the aspect ratio on the drag coefficient in stagnant water or glycerin solution. In each figure, aspect ratios vary widely (from 4 to 40), but CD cos φ with Re shows the same trend of variation. All of these data give the same indication that the drag coefficients of particles with different aspect ratios fall into a single curve of correlation. It is also noticed that the behavior of the drag coefficient for slender particles with a large aspect ratio is the same for particles with a small aspect ratio (r < 10). 5.4. Effect of Particle Materials. The use of synthetic fibers A and B and brass wire makes it possible to check the effect of the particle density on the drag coefficients under the same conditions. The results plotted in Figures 11 and 12 show that the relationship of CD versus Re is not a universal correlation and the particle density should be included in the desired general correlation. 5.5. Effect of Fluid Properties. Settling of synthetic fiber A in water and in a more viscous glycerin solution is shown in Figure 13 and that for synthetic fiber B in Figure 14. Drag coefficient data fall on two separate lines for water and for glycerin, respectively, showing distinct influences of liquid properties on the correlation. 6. General Correlation of the Drag Coefficient
Figure 6. Variation of orientation for a synthetic fiber A (d ) 0.160 mm) in flowing water.
correlated with the Reynolds number (Re), while the product of the drag coefficient and cosine of orientation (CD cos φ) displays a very regular correlation with Re. As shown typically in Figure 7, the product decreases
Drag coefficient CD of spherical particles is a dimensionless number, and eq 12 has fully encompassed the effects of the particle density, viscosity of liquid, and relative motion between particles and liquid via a single parameter Re. For spherical particles, CD could be uniquely expressed in terms of Re alone. However, for slender particles, the Reynolds number defined in terms of the volume-equivalent sphere diameter does not suffice to deduce a general correlation of the drag coefficient with reasonable accuracy. From dimensional analysis, the settling of slender particles is related to the Archimedes number Ar [Ar ) d3(Fp - Fl)Flg/µ2], drag coefficient CD, orientation φ, density ratio ∆F/Fl,
Figure 7. Comparison between stagnant and flowing liquid for synthetic fiber A (d ) 0.160 mm) in water.
7668
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004
Figure 8. Effect of the aspect ratio on the drag coefficient for synthetic fiber B (d ) 0.370 mm) in stagnant water.
Figure 11. Effect of the density on the drag coefficient in downward-flowing water.
Figure 12. Effect of the density on the drag coefficient in upwardflowing water. Figure 9. Effect of the aspect ratio on the drag coefficient for synthetic fiber B (d ) 0.370 mm) in stagnant glycerin.
Figure 13. Comparison between water and glycerin for synthetic fiber A (d ) 0.160 mm, l ) 3.0 mm) in downward flow. Figure 10. Effect of the aspect ratio on the drag coefficient for synthetic fiber A (d ) 0.160 mm) in stagnant water.
and aspect ratio l/d, in addition to the Reynolds number Re. Equations 8 and 9 suggest that the combined factor CD cos φ well correlates slender particle settling. The experiment also indicates that the variation of CD cos φ is independent of l/d. It is realized that the drag
coefficient of slender particles should be correlated with the Archimedes number and the density ratio included. Some correlating efforts suggested that the Archimedes number is necessary and the density ratio ∆F/Fp works better than ∆F/Fl. Because the Archimedes number has several dimensionally equivalent forms [the modified Archimedes number Ar* ) d3(Fp - Fl)2g/µ2 and Ar** ) d3Fl2g/µ2] and
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004 7669
Figure 14. Comparison between water and glycerin for synthetic fiber B (d ) 0.370 mm, l ) 9.2 mm) in upward flow.
Figure 16. Relationship between CD cos φ and Re in water and glycerin.
a function of the Reynolds number only. Besides, the combination of CD with cos φ sounds very effective, and the correlation of CD with cos φ as an independent variable is also confirmed to be unsuccessful. By comparison, it seems that the drag coefficient of slender particles is influenced by more factors than spherical particles owing to its special shape. 7. Conclusions
Figure 15. Relationship between CD cos φ and Re in water.
the density ratio may be expressed by (Fp - Fl)/Fp or Fp/ Fl, several ways of correlation were tried using different combinations of the Archimedes number and density ratio. It is found that the combination of the modified Archimedes number Ar* and Fp/Fl resulted in the best empirical equation:
CD cos φ )
24 (0.006983 + 0.6224Re-1.046) Re Fp -1.537 (Ar*)0.8524 (14) Fl
()
based on more than 1285 data points (for both accelerating and steady motion) in Figures 15 and 16. As shown in Figures 15 and 16, eq 14 correlates the data quite well in view of experimental error, and the product CD cos φ[Fp/Fl]1.537(Ar*)-0.8524 is independent of the particle aspect ratio, liquid viscosity, and velocity. The overall average relative error of the correlation with respect to the drag coefficient data is estimated to be 9.4%. Despite that the form of eq 14 is similar to that of eqs 12 and 13, a significant difference exists between the drag coefficients of spherical and slender particles. For instance, CD is obviously a function of the orientation, and both the Reynolds and modified Archimedes numbers appear in eq 14, while CD in eqs 12 and 13 is
Experiments are carried out on slender particles with a large aspect ratio in a laminar Newtonian fluid. Drag coefficients for particles with different densities and aspect ratios in accelerating and steady motion are determined in stagnant and flowing liquid of different viscosities. It is found that slender particles in Newtonian fluids will approach the stable horizontal orientation eventually. During settling, orientation will oscillate around the stable horizontal orientation regardless of the fluid velocity. In terms of the Reynolds number based on the slender particle cylinder diameter and relative velocity, a general correlation of the drag coefficient is established with the Archimedes number incorporated. The proposed correlation is of reasonable accuracy, reliable and suitable for a low-viscosity liquid and a particle density ranging from 1125 to 8000 kg‚m-3. The overall average relative error of the correlation is only 9.4% and is applicable to aspect ratios ranging between 4 and 50. Acknowledgment This work is supported financially by the National Natural Science Foundation of China (Nos. 20106016 and 20236050). Nomenclature Ar ) Archimedes number, Ar ) d3(Fp - Fl)Flg/µ2 Ar* ) modified Archimedes number, Ar* ) d3(Fp - Fl)2g/µ2 CD ) drag coefficient CDS ) drag coefficient of spherical particles Cvm) virtual mass coefficient d ) diameter of a spherical particle or a slender particle cylindrical diameter (m) F ) drag force (N) g ) acceleration of gravity (m‚s-2) l ) length of a slender particle (m)
7670
Ind. Eng. Chem. Res., Vol. 43, No. 23, 2004
r ) aspect ratio of a slender particle, r ) l/d Re ) Reynolds number, Re ) d|ul - up|Fl/µ S ) projected area of a particle normal to the direction of motion (m2) U ) average settling velocity of particles swarm (m‚s-1) U0 ) average settling velocity of a single slender particle (m‚s-1) up ) settling velocity of a particle (m‚s-1) ul ) velocity of fluid (m‚s-1) Vp ) volume of a slender particle, Vp ) πld2/4 (m3) Greek Letters Fl ) density of a fluid (kg‚m-3) Fp ) density of a particle (kg‚m-3) ψ ) sphericity of a particle φ ) volume concentration of particles in solution µ ) viscosity of a fluid (kg‚m-1‚s-1) φ ) angle between the axis of a slender particle and the horizon (deg)
Literature Cited (1) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops, and Particles; Academic Press: New York, 1978. (2) Liao, S. J. An analytic approximation of the drag coefficient for the viscous flow past a sphere. Int. J. Non-Linear Mech. 2002, 37, 1. (3) Saboni, A.; Alexandrova, S. Numerical study of the drag on a fluid sphere. AIChE J. 2002, 48, 2992.
(4) Liu, Y. J.; Joseph, D. D. Sedimentation of particles in polymer solutions. J. Fluid Mech. 1993, 255, 565. (5) Mckay, G.; Murphy, W. R.; Hillis, M. Settling characteristics of discs and cylinders. Chem. Eng. Res. Des. 1988, 66, 107. (6) Turney, M. A.; Cheumg, M. K.; Powell, R. L.; McCarthy, M. J. Hindered settling of rod-like particles measured with magnetic resonance imaging. AIChE J. 1995, 41, 251. (7) Haider, A. M.; Levenspiel, O. Drag coefficient and terminal velocity of spherical and non-spherical particles. Powder Technol. 1989, 58, 63. (8) Blumberg, P. N.; Mohr, C. M. Effect of orientation on the settling characteristics of cylindrical particles. AIChE J. 1968, 14, 331. (9) Kasper, G.; Niida, T.; Yang, M. Measurements of viscous drag on cylinders and chains of spheres with aspect ratios between 2 and 50. J. Aerosol Sci. 1985, 16, 535. (10) Unnikrishnan, A.; Chhabra, R. P. An experimental study of motion of cylinders in Newtonian fluids: wall effects and drag coefficient. Can. J. Chem. Eng. 1991, 69, 729. (11) Drew, D.; Cheng, L.; Lahey, R. T., Jr. The analysis of virtual mass effects in two-phase flow. Int. J. Multiphase Flow 1979, 5, 233. (12) Turton, R.; Levenspid, O. A short note on the drag correlation for spheres. Powder Technol. 1986, 47, 83. (13) Huang, P. Y.; Hu, H. H.; Joseph, D. D. Direct simulation of the sedimentation of elliptic particles in Oldroyd-B fluids. J. Fluid Mech. 1998, 362, 297.
Received for review June 15, 2004 Revised manuscript received August 10, 2004 Accepted August 24, 2004 IE049479K
