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New simple correlation formula for the drag coefficient of non-spherical particles
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Powder Technology 184 (2008) 361 – 365 www.elsevier.com/locate/powtec
Short communication
New simple correlation formula for the drag coefficient of non-spherical particles Andreas Hölzer, Martin Sommerfeld ⁎ Institut für Verfahrenstechnik, Fachbereich Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany Received 20 March 2007; received in revised form 31 July 2007; accepted 29 August 2007 Available online 5 September 2007
Abstract A simple correlation formula for the standard drag coefficient (i.e. a single stationary particle in a uniform flow) of arbitrary shaped particles is established using a large number of experimental data from the literature and a comprehensive numerical study [A. Hölzer, M. Sommerfeld, IUTAM Symposium on Computational Approaches to Multiphase Flow, Springer, 2006]. This new correlation formula accounts for the particle orientation over the entire range of Reynolds numbers up to the critical Reynolds number. Such a correlation may be easily used in the frame of Lagrangian computations where also the particle orientation along the trajectory is computed. © 2007 Elsevier B.V. All rights reserved. Keywords: Drag coefficient; Correlation; Orientation; Non-spherical particles
1. Introduction The motion of particles plays an important role in both technical and natural processes. Examples are combustion of pulverised coal, fibre suspension flow in paper forming, pneumatic conveying of granular materials, pollutant transport in the atmosphere and transport of sediment grains in rivers. Modelling of these processes relies mostly on the assumption of spherical particles. However, in practice, particles are non-spherical and the particle motion is affected by particle shape and particle orientation. For simulation of the motion of such particles detailed information on the drag acting on these particles are necessary, however, only averaged correlations of the drag coefficient for different particle shapes are available [1–3]. The characteristic dimension of a particle (d) which appears in the definitions of the Reynolds number (Re) and the drag coefficient (cD) has been defined as the diameter of the volumeequivalent sphere. Furthermore, the Reynolds number of a particle is based on the relative velocity of the fluid with respect to the particle (u) and the kinematic viscosity (υ): ud Re ¼ ð1Þ t ⁎ Corresponding author. E-mail address: [email protected] (M. Sommerfeld). 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.08.021
The drag coefficient of a particle is defined by: FD ; 2 p d2 q u 2 4
cD ¼ 1
ð2Þ
with FD being the drag force and ρ being the fluid density. 2. Literature review Analytical solutions for the drag of a particle only exist for spheres [4] and spheroids, including disks as special case [5,6], in the low Re number limit, the so-called Stokes region. The prolate spheroid with the symmetry axis parallel to the flow and with an axis ratio of 1.95 has the overall smallest drag coefficient. Batchelor [7] derived an approximate expression for the drag of cylinders with the symmetry axis both parallel and perpendicular to the flow, which is based on slender-body theory and thus valid for long cylinders only. There are many experimental studies on the drag of stationary particles in a uniform flow, mostly for spherical particles. From the beginning of the 20th century most of the experimental drag values were obtained by particle settling experiments at low Re numbers and by wind tunnel experiments at high Re numbers. Experimental cD values of spheres, disks and plates, lengthwise spheroids and streamline bodies,
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A. Hölzer, M. Sommerfeld / Powder Technology 184 (2008) 361–365
increases with decreasing sphericity. However, the sphericity cannot describe the influence of the particle orientation on cD, since the sphericity is independent of the orientation. Furthermore, some spheroids experience a smaller drag than a sphere in the Stokes region [6] and lengthwise plates, spheroids and streamline bodies experience a smaller drag than a sphere at high Re numbers [19,27,28,31], which also cannot be described by the sphericity as parameter. In the Stokes region, theoretical (sphere [4], spheroids [6], cylinders [7]) and experimental (e.g. [21–23]) investigations indicate that cD of all bodies decreases inversely proportional to the Re number. Leith [33] suggests the following equation for cD in the Stokes region: cD ¼ Fig. 1. Drag coefficient of differently shaped particles from the literature as a , Stokes; , Sphere [8–16]; ♢, Disks and function of the Reynolds number. plates [14–20]; ✕, Isometric bodies [21–23]; |, Minerals [24–26]; ◯, Spheroids and streamline bodies [27–31].
isometric particles such as cubes, tetrahedrons and octahedrons and irregularly shaped particles such as minerals are summarised in Fig. 1. One can see the strong influence of the Re number, particle shape and particle orientation. Crosswise disks and plates experience a large drag over the entire range of Re numbers. Lengthwise disks and plates experience a large drag at low Re numbers, but a very small drag at high Re numbers. At high Re numbers, lengthwise spheroids and streamline bodies have a very small drag too. Isometric bodies and minerals have a small drag at low Re numbers and a small or an intermediate drag at high Re numbers. At low Re numbers, lengthwise spheroids experience the smallest drag. Reliable correlations for cD exist for the sphere for the entire range of subcritical Re numbers [1,32]. For non-spherical particles, Haider und Levenspiel [1] established a correlation formula for cD from experimental data which contains the Re number and the sphericity (Φ) as parameters. No restrictions of applicability in terms of Re and Φ are given for this correlation. The sphericity is the ratio between the surface area of the volume equivalent sphere and the surface area of the considered particle. In Haider und Levenspiel's correlation the value of cD
8 1 16 1 pffiffiffiffiffiffiffi þ pffiffiffiffi Re U8 Re U
ð3Þ
Analogous to the sphericity, the crosswise sphericity (Φ⊥) is the ratio between the cross-sectional area of the volume equivalent sphere and the projected cross-sectional area of the considered particle perpendicular to the flow. The first term in Eq. (3) stands for the pressure or form drag, associated with the size of the projected cross-sectional area, and the second term stands for the friction drag, associated with the size of the surface area. Ganser [2] established a correlation formula which contains Leith's Eq. (3) for the Stokes region and the equation: cD ¼ 0:42d 101:8148ð
log UÞ0:5743
ð4Þ
for the Newton region, i.e. at very high subcritical Re numbers where cD is approximately constant. No validation limits for the parameters of Eqs. (3) and (4) were given by the authors. For all isometric and crosswise oriented particles in the Newton regime cD is almost solely determined by form drag according to experimental results (e.g. [14,16,23,34]) and Newton's theoretical investigations. According to numerical simulations [35], the drag of isometric particles is approximately proportional to the size of the projected cross-sectional area or the reciprocal crosswise sphericity at high Re numbers. Tran-Cong et al. [3] determined the same trend from settling experiments with agglomerates of ordered packed spheres and included this relationship in their proposed correlation formula,
Table 1 Mean (upper table) and maximum (bottom table) relative deviations between the correlation formulas of Haider and Levenspiel [1], Ganser including Leith [2] and the present formulas (Eqs. (9) and (10)) and experimental values (references see captions of Figs. 2, 3 and 4) Haider and Levenspiel [1]
Ganser incl. Leith [2]
Present Eq. (9)
Present Eq. (10)
Mean relative deviation Sphere (683 values) Isometric particles (655 values) Cuboids and cylinders (337 values) Disks and plates (386 values) All Data (2061 values)
6.59% 6.65% 42.3% 2 103% 383%
10.9% 6.46% 38.4% 1.8 103% 348%
9.17% 10.5% 27.2% 17.7% 14.1%
9.17% 10.9% 29% 16.8% 14.4%
Maximum relative deviation Sphere (683 values) Isometric particles (655 values) Cuboids and cylinders (337 values) Disks and plates (386 values)
44% 50% 1.1 103% 2.1 104%
43% 55% 1.1 103% 2.4 104%
45% 68% 88% 75%
45% 68% 88% 75%
A. Hölzer, M. Sommerfeld / Powder Technology 184 (2008) 361–365
363
which is however only valid for Re numbers up to 1500 and a certain range of crosswise sphericities. Lengthwise disks and plates have a very small cross-sectional area and a very large surface area. Thus, the friction drag is important for them at high Re numbers also. In addition the cD value decreases with the Re number at high Re numbers, which is mathematically described by the following equation [36]: cD ¼ 1:327 2
14 1 8 depth 4 1 1 1 pffiffiffiffiffiffi p4 9 length U34 Re
ð5Þ
for lengthwise plates in a laminar flow, which can be reduced to: cD ¼ 3:43
1
1 pffiffiffiffiffiffi U Re 3 4
ð6Þ
for square plates.
Fig. 3. Comparison between the correlated and experimental drag coefficients of cuboids and cylinders as a function of the Reynolds number [16,22,38,43,45,47–57].
3. Results and discussion The use of the lengthwise sphericity (Φ||) instead of the crosswise sphericity in Leith's Eq. (3) for cD in the Stokes region leads to a better approximation of cD as a function of the particle orientation. The lengthwise sphericity is the ratio between the cross-sectional area of the volume equivalent sphere and the difference between half the surface area and the mean longitudinal (i.e. parallel to the direction of relative flow) projected cross-sectional area of the considered particle. It must be taken into account that the longitudinal cross-sectional area of a particle has in general, depending on the angle of view, different values. The mean projected longitudinal cross-sectional area is therefore the arithmetic average over an entire revolution of the particle around the axis parallel to the direction of relative flow. The new correlation for the Stokes region reads: 8 1 16 1 pffiffiffiffiffiffi þ pffiffiffiffi cD ¼ ð7Þ Re Ujj Re U For a sphere (Φ = Φ|| = 1), this correlation agrees with Stokes' analytical solution [4]. For the crosswise and lengthwise disk
Fig. 2. Comparison between the correlated and experimental drag coefficients of spheres [8–16,21–23,37–44] and isometric particles [21–23,45,46] as a function of the Reynolds number.
[6], Eq. (7) predicts a cD which is 5% too small and 25% too large, respectively. The best correlation for cD at high Re numbers consists of a term according to Eq. (6) and of a term which is proportional to the size of the projected cross-sectional area according to Tran-Cong et al. [3] with a factor of proportionality analogous to Eq. (4): 3 1 cD ¼ pffiffiffiffiffiffi 3 þ 0:42100:4ð Re U4
log UÞ0:2
1 U8
ð8Þ
The correlation formula for cD over the entire range of Re numbers results from the addition of the Eqs. (7) and (8): cD ¼
8 1 16 1 3 1 pffiffiffiffi þ pffiffiffiffiffiffi 3 þ 0:42100:4ð pffiffiffiffiffiffi þ Re Ujj Re U Re U4
log UÞ0:2
1 U8 ð9Þ
Table 1 shows the mean and maximum relative deviations |cDcorr − cDexp |/cDexp between experimental values and the
Fig. 4. Comparison between the correlated and experimental drag coefficients of disks and plates as a function of the Reynolds number [14– 20,34,43,45,48,54,58].
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correlation formulas of Haider and Levenspiel [1], Ganser [2] including Leith's Eq. (3) for the Stokes region and the present formula (Eq. (9)). It should be mentioned that all experimental data have a certain scatter and some data points might be imprecise. Therefore, the true deviations are supposed to be smaller than the deviations listed in Table 1. The present correlation (Eq. (9)) is plotted in comparison to experimental cD values of spheres and isometric particles in Fig. 2. For spheres and isometric particles, the correlations of Haider and Levenspiel and of Ganser predict cD more exactly than the present correlation; e.g. Haider and Levenspiel's and Ganser's correlations describe the minimum in cD at intermediate Re numbers due to the more sophisticated form of their equations. Fig. 3 shows the present correlation (Eq. (9)) for cuboids and cylinders compared to experimental results and Fig. 4 shows the present correlation (Eq. (9)) of lengthwise and crosswise disks and plates compared to experimental results. The new correlation considers the orientation of the particles over the entire range of Re numbers. For lengthwise disks and plates, it predicts a large cD at low Re numbers and a very small cD at high Re numbers. In contrast, the formulas of Haider and Levenspiel and of Ganser do not consider the particle orientation at high Re numbers and, thus, their predicted cD of lengthwise disks and plates is up to three orders of magnitude too large at high Re numbers. This is to be recognized by the very large relative deviation of their values for disks and plates in Table 1. The present correlation (Eq. (9)) has with 14.1% by far the smallest mean relative deviation of all data considered, compared to the deviations of 383% and 348% for the correlations of Haider and Levenspiel and Ganser, respectively (Table 1). Thus, the present correlation (Eq. (9)) is, to the author's knowledge, the most exact available correlation for cD of arbitrary shaped particles. The Leith correlation (Eq. (3)) also is a good approximation for cD in the Stokes region. However, it describes the dependence of cD on the orientation rather poorly, which is on the other hand relative unimportant at low Re numbers. The advantage of Eq. (3) compared to Eq. (7) is, that the crosswise sphericity in Eq. (3) is more easily to determine than the lengthwise sphericity in Eq. (7). Doing so in correlation Eq. (9), the lengthwise sphericity would be substituted by the crosswise sphericity, yielding:
cD ¼
8 1 16 1 3 1 pffiffiffiffiffiffiffi þ pffiffiffiffi þ pffiffiffiffiffiffi 3 þ 0:42100:4ð Re U8 Re U Re U4
log UÞ0:2
1 U8 ð10Þ
4. Conclusions On the basis of experimental data from the literature and of a numerical study [35], the correlation formula for the drag coefficient: cD ¼
log UÞ0:2
1 U8
was established, which depends on the shape, the orientation and the Reynolds number (Re) of the particle. The sphericity (Φ) represents the ratio between the surface area of the volume equivalent sphere and that of the considered particle, the crosswise sphericity (Φ⊥) is the ratio between the cross-sectional area of the volume equivalent sphere and the projected crosssectional area of the considered particle and the lengthwise sphericity (Φ||) is the ratio between the cross-sectional area of the volume equivalent sphere and the difference between half the surface area and the mean projected longitudinal crosssectional area of the considered particle. The correlation formula established considers the particle orientation over the entire range of Reynolds numbers up to the critical Reynolds number and has no known limits of applicability in terms of Φ, Φ⊥ and Φ||. The mean relative deviation between the new correlation and 2061 experimental data from the literature for differently shaped particles is 14.1% and is the smallest deviation of all correlation formulas known to the author. If in the new correlation formula established the lengthwise sphericity is substituted by the more easily determinable crosswise sphericity, the deviation becomes 14.4%. Thus, this simplified formula can also be applied with high confidence in numerical computations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
The relative deviations between this correlation formula (Eq. (10)) and the experimental values are listed in the last column of Table 1. The mean relative deviation of all data is with 14.4% only slightly larger than the mean deviation of the correlation Eq. (9) with 14.1%. Thus, the simplified formula (Eq. (10)) has a small confidence interval too.
8 1 16 1 3 1 pffiffiffiffi þ pffiffiffiffiffiffi 3 þ 0:42100:4ð pffiffiffiffiffiffi þ Re Ujj Re U Re U4
[18] [19] [20] [21] [22]
A. Haider, O. Levenspiel, Powder Technol. 58 (1989) 63. G.H. Ganser, Powder Technol. 77 (1993) 143. S. Tran-Cong, M. Hay, E.E. Michaelides, Powder Technol. 139 (2004) 21. G.G. Stokes, Camb. Phil. Trans. 9 (1851) 8. A. Oberbeck, J. Reine Angew. Math. 81 (1876) 62. C.W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Akademische Verlagsgesellschaft, Leipzig, 1927. G.K. Batchelor, J. Fluid Mech. 44 (1970) 419. H.S. Allen, Phil. Mag. Ser. 5 50 (1900) 323. H.D. Arnold, Phil. Mag. Ser. 6 22 (1911) 755. D.L. Bacon, E.G. Reid, NACA Report No. 185, 1924. G. Eiffel, CR Hebd. Séances Acad. Sci. 155 (1912) 1597. E.N. Jacobs, NACA Technical Notes No. 312, 1929. H. Liebster, Ann. Phys. 82 (1927) 541. F.W. Roos, W.W. Willmarth, AIAA J. 9 (1971) 285. J. Schmiedel, Physik. Z. 29 (1928) 593. C. Wieselsberger, Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen, 2, 1923, p. 22. O. Föppl, Windkräfte an ebenen und gewölbten Platten. Dissertation, Springer, Berlin, 1911. L. Squires, W. Squires, Trans. Am. Inst. Chem. Eng. 33 (1937) 1. C. Wieselsberger, Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen, 1, 1923, p. 120. W.W. Willmarth, N.E. Hawk, R.L. Harvey, Phys. Fluids 7 (1964) 197. K.C.R. Chowdhury, W. Fritz, Chem. Eng. Sci. 11 (1959) 92. J.S. McNown, J. Malaika, Trans. AGU 31 (1950) 74.
A. Hölzer, M. Sommerfeld / Powder Technology 184 (2008) 361–365 [23] E.S. Pettyjohn, E.B. Christiansen, Chem. Eng. Prog. 44 (1948) 157. [24] H. Heywood, Proc. Inst. Mech. Eng. 140 (1938) 257. [25] J.D. Hottovy, N.D. Sylvester, Ind. Eng. Chem. Process Des. Dev. 18 (1979) 433. [26] L.W. Needham, N.W. Hill, Fuel Sci. Pract. 26 (1947) 101. [27] I.H. Abbott, NACA Report No. 451, 1934. [28] H.L. Dryden, A.M. Kuethe, NACA Technical Report No. 342, 1931. [29] H.A. Dwyer, D.S. Dandy, Phys. Fluids A 2 (1990) 2110. [30] J.H. Masliyah, N. Epstein, J. Fluid Mech. 44 (1970) 493. [31] C. Wieselsberger, ZFM, Z. Flugtech. Mot.luftschiffahrt 5 (1914) 140. [32] R. Turton, O. Levenspiel, Powder Technol. 47 (1986) 83. [33] D. Leith, Aerosol Sci. Tech. 6 (1987) 153. [34] G.B. Schubauer, H.L. Dryden, NACA Report No. 546, 1937. [35] A. Hölzer, M. Sommerfeld, in: S. Balachandar, A. Prosperetti (Eds.), IUTAM Symposium on Computational Approaches to Disperse Multiphase Flow, 99-108, Springer Dordrecht, 2006. [36] H. Blasius, Z. Math. Phys. 56 (1908) 1. [37] E. Achenbach, J. Fluid Mech. 54 (1972) 565. [38] E.B. Christiansen, D.H. Barker, AIChE J. 11 (1965) 145. [39] O. Flachsbart, Physik. Z. 28 (1927) 461. [40] R.J. Gibbs, M.D. Matthews, D.A. Link, J. Sediment. Petrol. 41 (1971) 7. [41] R.G. Lunnon, Proc. R. Soc. Lond., A 118 (1928) 680.
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[47] [48]
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W. Möller, Physik. Z. 39 (1938) 57. G.P. Williams, J. Sediment. Petrol. 36 (1966) 255. J. Zeleny, L.W. McKeehan, Phys. Rev. 30 (1910) 535. S. Gurel, S.G. Ward, R.L. Whitmore, Brit. J. Appl. Phys. 6 (1955) 83. P. Schulz, Neue Bestimmungen der Konstanten der Fallgesetze in der nassen Aufbereitung mit Hilfe der Kinematographie und Betrachtungen über das Gleichfälligkeitsgesetz. Dissertation, Noske, Leipzig, 1914. H.A. Becker, Can. J. Chem. Eng. 37 (1959) 85. G. Eiffel, in: H. Dunod, E. Pinat (Eds.), La résistance de l'air et de l'aviation expériences effectuées au laboratoire du Champ-de-Mars, Paris, 1910. J.F. Heiss, J. Coull, Chem. Eng. Prog. 48 (1952) 133. G. Kasper, T. Niida, M. Yang, J. Aerosol Sci. 16 (1985) 535. P.D. Komar, J. Geol. 88 (1980) 327. I.A. Lasso, P.D. Weidman, Phys. Fluids 29 (1986) 3921. P.S. Lee, T. Niida, D.T. Shaw, Aerosol Sci. Tech. 1 (1982) 259. J. Ludwig, Chem.-Ztg. 22 (1955) 774. E.K. Marchildon, W.H. Gauvin, AIChE J. 25 (1964) 938. G. McKay, W.R. Murphy, M. Hillis, Chem. Eng. Res. Des. 66 (1988) 107. A.W. Sheaffer, J. Aerosol Sci. 18 (1987) 11. H. Blasius, Mitt. Forschungsarb. VDI 131 (1913) 1.
Powder Technology 184 (2008) 361 – 365 www.elsevier.com/locate/powtec
Short communication
New simple correlation formula for the drag coefficient of non-spherical particles Andreas Hölzer, Martin Sommerfeld ⁎ Institut für Verfahrenstechnik, Fachbereich Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany Received 20 March 2007; received in revised form 31 July 2007; accepted 29 August 2007 Available online 5 September 2007
Abstract A simple correlation formula for the standard drag coefficient (i.e. a single stationary particle in a uniform flow) of arbitrary shaped particles is established using a large number of experimental data from the literature and a comprehensive numerical study [A. Hölzer, M. Sommerfeld, IUTAM Symposium on Computational Approaches to Multiphase Flow, Springer, 2006]. This new correlation formula accounts for the particle orientation over the entire range of Reynolds numbers up to the critical Reynolds number. Such a correlation may be easily used in the frame of Lagrangian computations where also the particle orientation along the trajectory is computed. © 2007 Elsevier B.V. All rights reserved. Keywords: Drag coefficient; Correlation; Orientation; Non-spherical particles
1. Introduction The motion of particles plays an important role in both technical and natural processes. Examples are combustion of pulverised coal, fibre suspension flow in paper forming, pneumatic conveying of granular materials, pollutant transport in the atmosphere and transport of sediment grains in rivers. Modelling of these processes relies mostly on the assumption of spherical particles. However, in practice, particles are non-spherical and the particle motion is affected by particle shape and particle orientation. For simulation of the motion of such particles detailed information on the drag acting on these particles are necessary, however, only averaged correlations of the drag coefficient for different particle shapes are available [1–3]. The characteristic dimension of a particle (d) which appears in the definitions of the Reynolds number (Re) and the drag coefficient (cD) has been defined as the diameter of the volumeequivalent sphere. Furthermore, the Reynolds number of a particle is based on the relative velocity of the fluid with respect to the particle (u) and the kinematic viscosity (υ): ud Re ¼ ð1Þ t ⁎ Corresponding author. E-mail address: [email protected] (M. Sommerfeld). 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.08.021
The drag coefficient of a particle is defined by: FD ; 2 p d2 q u 2 4
cD ¼ 1
ð2Þ
with FD being the drag force and ρ being the fluid density. 2. Literature review Analytical solutions for the drag of a particle only exist for spheres [4] and spheroids, including disks as special case [5,6], in the low Re number limit, the so-called Stokes region. The prolate spheroid with the symmetry axis parallel to the flow and with an axis ratio of 1.95 has the overall smallest drag coefficient. Batchelor [7] derived an approximate expression for the drag of cylinders with the symmetry axis both parallel and perpendicular to the flow, which is based on slender-body theory and thus valid for long cylinders only. There are many experimental studies on the drag of stationary particles in a uniform flow, mostly for spherical particles. From the beginning of the 20th century most of the experimental drag values were obtained by particle settling experiments at low Re numbers and by wind tunnel experiments at high Re numbers. Experimental cD values of spheres, disks and plates, lengthwise spheroids and streamline bodies,
362
A. Hölzer, M. Sommerfeld / Powder Technology 184 (2008) 361–365
increases with decreasing sphericity. However, the sphericity cannot describe the influence of the particle orientation on cD, since the sphericity is independent of the orientation. Furthermore, some spheroids experience a smaller drag than a sphere in the Stokes region [6] and lengthwise plates, spheroids and streamline bodies experience a smaller drag than a sphere at high Re numbers [19,27,28,31], which also cannot be described by the sphericity as parameter. In the Stokes region, theoretical (sphere [4], spheroids [6], cylinders [7]) and experimental (e.g. [21–23]) investigations indicate that cD of all bodies decreases inversely proportional to the Re number. Leith [33] suggests the following equation for cD in the Stokes region: cD ¼ Fig. 1. Drag coefficient of differently shaped particles from the literature as a , Stokes; , Sphere [8–16]; ♢, Disks and function of the Reynolds number. plates [14–20]; ✕, Isometric bodies [21–23]; |, Minerals [24–26]; ◯, Spheroids and streamline bodies [27–31].
isometric particles such as cubes, tetrahedrons and octahedrons and irregularly shaped particles such as minerals are summarised in Fig. 1. One can see the strong influence of the Re number, particle shape and particle orientation. Crosswise disks and plates experience a large drag over the entire range of Re numbers. Lengthwise disks and plates experience a large drag at low Re numbers, but a very small drag at high Re numbers. At high Re numbers, lengthwise spheroids and streamline bodies have a very small drag too. Isometric bodies and minerals have a small drag at low Re numbers and a small or an intermediate drag at high Re numbers. At low Re numbers, lengthwise spheroids experience the smallest drag. Reliable correlations for cD exist for the sphere for the entire range of subcritical Re numbers [1,32]. For non-spherical particles, Haider und Levenspiel [1] established a correlation formula for cD from experimental data which contains the Re number and the sphericity (Φ) as parameters. No restrictions of applicability in terms of Re and Φ are given for this correlation. The sphericity is the ratio between the surface area of the volume equivalent sphere and the surface area of the considered particle. In Haider und Levenspiel's correlation the value of cD
8 1 16 1 pffiffiffiffiffiffiffi þ pffiffiffiffi Re U8 Re U
ð3Þ
Analogous to the sphericity, the crosswise sphericity (Φ⊥) is the ratio between the cross-sectional area of the volume equivalent sphere and the projected cross-sectional area of the considered particle perpendicular to the flow. The first term in Eq. (3) stands for the pressure or form drag, associated with the size of the projected cross-sectional area, and the second term stands for the friction drag, associated with the size of the surface area. Ganser [2] established a correlation formula which contains Leith's Eq. (3) for the Stokes region and the equation: cD ¼ 0:42d 101:8148ð
log UÞ0:5743
ð4Þ
for the Newton region, i.e. at very high subcritical Re numbers where cD is approximately constant. No validation limits for the parameters of Eqs. (3) and (4) were given by the authors. For all isometric and crosswise oriented particles in the Newton regime cD is almost solely determined by form drag according to experimental results (e.g. [14,16,23,34]) and Newton's theoretical investigations. According to numerical simulations [35], the drag of isometric particles is approximately proportional to the size of the projected cross-sectional area or the reciprocal crosswise sphericity at high Re numbers. Tran-Cong et al. [3] determined the same trend from settling experiments with agglomerates of ordered packed spheres and included this relationship in their proposed correlation formula,
Table 1 Mean (upper table) and maximum (bottom table) relative deviations between the correlation formulas of Haider and Levenspiel [1], Ganser including Leith [2] and the present formulas (Eqs. (9) and (10)) and experimental values (references see captions of Figs. 2, 3 and 4) Haider and Levenspiel [1]
Ganser incl. Leith [2]
Present Eq. (9)
Present Eq. (10)
Mean relative deviation Sphere (683 values) Isometric particles (655 values) Cuboids and cylinders (337 values) Disks and plates (386 values) All Data (2061 values)
6.59% 6.65% 42.3% 2 103% 383%
10.9% 6.46% 38.4% 1.8 103% 348%
9.17% 10.5% 27.2% 17.7% 14.1%
9.17% 10.9% 29% 16.8% 14.4%
Maximum relative deviation Sphere (683 values) Isometric particles (655 values) Cuboids and cylinders (337 values) Disks and plates (386 values)
44% 50% 1.1 103% 2.1 104%
43% 55% 1.1 103% 2.4 104%
45% 68% 88% 75%
45% 68% 88% 75%
A. Hölzer, M. Sommerfeld / Powder Technology 184 (2008) 361–365
363
which is however only valid for Re numbers up to 1500 and a certain range of crosswise sphericities. Lengthwise disks and plates have a very small cross-sectional area and a very large surface area. Thus, the friction drag is important for them at high Re numbers also. In addition the cD value decreases with the Re number at high Re numbers, which is mathematically described by the following equation [36]: cD ¼ 1:327 2
14 1 8 depth 4 1 1 1 pffiffiffiffiffiffi p4 9 length U34 Re
ð5Þ
for lengthwise plates in a laminar flow, which can be reduced to: cD ¼ 3:43
1
1 pffiffiffiffiffiffi U Re 3 4
ð6Þ
for square plates.
Fig. 3. Comparison between the correlated and experimental drag coefficients of cuboids and cylinders as a function of the Reynolds number [16,22,38,43,45,47–57].
3. Results and discussion The use of the lengthwise sphericity (Φ||) instead of the crosswise sphericity in Leith's Eq. (3) for cD in the Stokes region leads to a better approximation of cD as a function of the particle orientation. The lengthwise sphericity is the ratio between the cross-sectional area of the volume equivalent sphere and the difference between half the surface area and the mean longitudinal (i.e. parallel to the direction of relative flow) projected cross-sectional area of the considered particle. It must be taken into account that the longitudinal cross-sectional area of a particle has in general, depending on the angle of view, different values. The mean projected longitudinal cross-sectional area is therefore the arithmetic average over an entire revolution of the particle around the axis parallel to the direction of relative flow. The new correlation for the Stokes region reads: 8 1 16 1 pffiffiffiffiffiffi þ pffiffiffiffi cD ¼ ð7Þ Re Ujj Re U For a sphere (Φ = Φ|| = 1), this correlation agrees with Stokes' analytical solution [4]. For the crosswise and lengthwise disk
Fig. 2. Comparison between the correlated and experimental drag coefficients of spheres [8–16,21–23,37–44] and isometric particles [21–23,45,46] as a function of the Reynolds number.
[6], Eq. (7) predicts a cD which is 5% too small and 25% too large, respectively. The best correlation for cD at high Re numbers consists of a term according to Eq. (6) and of a term which is proportional to the size of the projected cross-sectional area according to Tran-Cong et al. [3] with a factor of proportionality analogous to Eq. (4): 3 1 cD ¼ pffiffiffiffiffiffi 3 þ 0:42100:4ð Re U4
log UÞ0:2
1 U8
ð8Þ
The correlation formula for cD over the entire range of Re numbers results from the addition of the Eqs. (7) and (8): cD ¼
8 1 16 1 3 1 pffiffiffiffi þ pffiffiffiffiffiffi 3 þ 0:42100:4ð pffiffiffiffiffiffi þ Re Ujj Re U Re U4
log UÞ0:2
1 U8 ð9Þ
Table 1 shows the mean and maximum relative deviations |cDcorr − cDexp |/cDexp between experimental values and the
Fig. 4. Comparison between the correlated and experimental drag coefficients of disks and plates as a function of the Reynolds number [14– 20,34,43,45,48,54,58].
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A. Hölzer, M. Sommerfeld / Powder Technology 184 (2008) 361–365
correlation formulas of Haider and Levenspiel [1], Ganser [2] including Leith's Eq. (3) for the Stokes region and the present formula (Eq. (9)). It should be mentioned that all experimental data have a certain scatter and some data points might be imprecise. Therefore, the true deviations are supposed to be smaller than the deviations listed in Table 1. The present correlation (Eq. (9)) is plotted in comparison to experimental cD values of spheres and isometric particles in Fig. 2. For spheres and isometric particles, the correlations of Haider and Levenspiel and of Ganser predict cD more exactly than the present correlation; e.g. Haider and Levenspiel's and Ganser's correlations describe the minimum in cD at intermediate Re numbers due to the more sophisticated form of their equations. Fig. 3 shows the present correlation (Eq. (9)) for cuboids and cylinders compared to experimental results and Fig. 4 shows the present correlation (Eq. (9)) of lengthwise and crosswise disks and plates compared to experimental results. The new correlation considers the orientation of the particles over the entire range of Re numbers. For lengthwise disks and plates, it predicts a large cD at low Re numbers and a very small cD at high Re numbers. In contrast, the formulas of Haider and Levenspiel and of Ganser do not consider the particle orientation at high Re numbers and, thus, their predicted cD of lengthwise disks and plates is up to three orders of magnitude too large at high Re numbers. This is to be recognized by the very large relative deviation of their values for disks and plates in Table 1. The present correlation (Eq. (9)) has with 14.1% by far the smallest mean relative deviation of all data considered, compared to the deviations of 383% and 348% for the correlations of Haider and Levenspiel and Ganser, respectively (Table 1). Thus, the present correlation (Eq. (9)) is, to the author's knowledge, the most exact available correlation for cD of arbitrary shaped particles. The Leith correlation (Eq. (3)) also is a good approximation for cD in the Stokes region. However, it describes the dependence of cD on the orientation rather poorly, which is on the other hand relative unimportant at low Re numbers. The advantage of Eq. (3) compared to Eq. (7) is, that the crosswise sphericity in Eq. (3) is more easily to determine than the lengthwise sphericity in Eq. (7). Doing so in correlation Eq. (9), the lengthwise sphericity would be substituted by the crosswise sphericity, yielding:
cD ¼
8 1 16 1 3 1 pffiffiffiffiffiffiffi þ pffiffiffiffi þ pffiffiffiffiffiffi 3 þ 0:42100:4ð Re U8 Re U Re U4
log UÞ0:2
1 U8 ð10Þ
4. Conclusions On the basis of experimental data from the literature and of a numerical study [35], the correlation formula for the drag coefficient: cD ¼
log UÞ0:2
1 U8
was established, which depends on the shape, the orientation and the Reynolds number (Re) of the particle. The sphericity (Φ) represents the ratio between the surface area of the volume equivalent sphere and that of the considered particle, the crosswise sphericity (Φ⊥) is the ratio between the cross-sectional area of the volume equivalent sphere and the projected crosssectional area of the considered particle and the lengthwise sphericity (Φ||) is the ratio between the cross-sectional area of the volume equivalent sphere and the difference between half the surface area and the mean projected longitudinal crosssectional area of the considered particle. The correlation formula established considers the particle orientation over the entire range of Reynolds numbers up to the critical Reynolds number and has no known limits of applicability in terms of Φ, Φ⊥ and Φ||. The mean relative deviation between the new correlation and 2061 experimental data from the literature for differently shaped particles is 14.1% and is the smallest deviation of all correlation formulas known to the author. If in the new correlation formula established the lengthwise sphericity is substituted by the more easily determinable crosswise sphericity, the deviation becomes 14.4%. Thus, this simplified formula can also be applied with high confidence in numerical computations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
The relative deviations between this correlation formula (Eq. (10)) and the experimental values are listed in the last column of Table 1. The mean relative deviation of all data is with 14.4% only slightly larger than the mean deviation of the correlation Eq. (9) with 14.1%. Thus, the simplified formula (Eq. (10)) has a small confidence interval too.
8 1 16 1 3 1 pffiffiffiffi þ pffiffiffiffiffiffi 3 þ 0:42100:4ð pffiffiffiffiffiffi þ Re Ujj Re U Re U4
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