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Drag coefficient of flow around a sphere: Matching asymptotically the wide trend
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Powder Technology 186 (2008) 218 – 223 www.elsevier.com/locate/powtec
Drag coefficient of flow around a sphere: Matching asymptotically the wide trend Jaber Almedeij ⁎ Civil Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 3 May 2007; received in revised form 7 December 2007; accepted 10 December 2007 Available online 21 February 2008
Abstract An empirical relationship of drag coefficient of flow around a sphere is developed for the entire range of Particle Reynolds numbers reported in the literature from Stokes regime to the condition when turbulent boundary layer prevails. The relationship is obtained using an approach to match asymptotically the wide trend of drag coefficient. The matching approach, which relies on dividing the wide trend into smaller segments that can be combined into an overall relationship, employs regression techniques and thus warrants the best-fit accuracy results. The relationship is calibrated with experimental data available in the literature covering the entire range for Reynolds numbers up to ~ 106. For Reynolds values greater than 106, the relationship renders a drag coefficient of 0.2. The performance of the relationship is tested and compared with other suitable models found in the literature. This relationship is also transformed into an explicit expression for settling velocity calculations. © 2008 Elsevier B.V. All rights reserved. Keywords: Regression; Reynolds number; Settling velocity; Turbulent boundary; Viscous flow
1. Introduction For the case of flow around a sphere, certain hydraulic analyses require determining the drag coefficient as a function of Particle Reynolds number. This is, for example, worthy to estimate the particle settling velocity, which is a parameter required for the diverse implications of sediment transport and deposition in pipelines (e.g. [1–4]) and alluvial channels (e.g. [5–7]), and of chemical and powder processes (e.g. [8,9]). However, most research efforts report existing difficulties to model theoretically the relationship of drag coefficient. One problem is that the drag coefficient cannot be expressed in an analytical form for a wide range of Particle Reynolds numbers, because the flow condition during the process is highly complicated [10]. This relationship though can be provided in charts and tables [12], determined experimentally by observing the settling velocities in still fluids or by measuring the drag of spheres in wind and tunnels [11]. Owing to the high advances in the development of computer and software applications, the numerical ⁎ Tel.: +965 4987603; fax: +965 4817524. E-mail address: [email protected]. 0032-5910/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.12.006
data in charts and tables representing this relationship would not be practical for the fast computational schemes. Rather, a numerical expression would be necessary. Attempts have thus been made to express the relationship empirically in order to extend the range of prediction. Until now, most of the empirical expressions are not satisfactory [13,14]. Only few empirical attempts, although they are valid for restricted ranges of Reynolds numbers, present acceptable drag coefficient results (e.g. [15,16]). The aim of the study is to derive an empirical relationship of drag coefficient of flow around a sphere valid for all Particle Reynolds numbers. Initially, the relationship is derived by using an approach to match asymptotically the wide trend of drag coefficient. Then, the accuracy of the relationship is tested and compared with other suitable models in the literature. The relationship is then transformed into an explicit expression of settling velocity as a function of particle diameter. 2. Theory and proposed relationships A graphical presentation of the experimental data of drag cofficient, CD, as a function of Particle Reynolds number, Re = wd/v, found in the vast literature is that given by Rouse [17].
J. Almedeij / Powder Technology 186 (2008) 218–223
219
Table 1 Summary of some empirical relationships, range of applicability, and accuracy Author Schiller et al. [22] Torobin et al. [23] Flemmer and Banks [16]
Turton and Levenspiel [15] Hesketh et al. [24]
Empirical relationship 0:687 24 CD ¼ Re 1 þ 0:15Re 0:63 24 þ 0:0026Re1:38 CD ¼ Re 1 þ 0:197Re E 24 CD ¼ Re 10 where E ¼ 0:261Re0:369 0:105Re0:431 1þð0:124 log ReÞ2 24 0:413 CD ¼ Re 1 þ 0:173Re0:657 þ 1þ16300Re 1:09 h i2 CD ¼ 0:352 þ ð0:124 þ 24=ReÞ1=2
5
a b
MASE a
2 b Re b 800 1 b Re b 100 Re b 3 × 105
1.03102 1.14055 1.03538 b
Re b 2 × 105
1.03391 b
1=4 0.1 b Re b 104 CD ¼ 1 0:5eð0:182Þ þ 10:11Re2=3 eð0:952Re Þ 0:125104 ReÞ 4 4=3 ð1:3Re1=2 Þ ð 0:03859Re e þ 0:037 10 Re e
0:116 1010 Re2 eð0:44410
Ceylan et al. [25] Almedeij
Range of applicability
1.09531
ReÞ
0.1 b Re b 106 Re b 106
Eq. (9)
1.08306 1.02874 b
Obtained using Eq. (10) and for the range of experimental data within which the model is valid. Calculated from experimental data starting from Re = 0.0001.
As the particle size or flow velocity increases for a given Kinematic viscosity, so does the Reynolds number, and then the character of flow changes. For very small Reynolds numbers, Stokes proposed an analytical solution of drag coefficient by solving the general differential equation of Navier–Stokes CD ¼
24 Re
ð1Þ
The Stokes solution neglects the effects of inertia and is acceptable roughly for Re b 0.4, when the laminar boundary layer is not separated from the particle. This solution reflects entirely the viscous effect of flow. For larger Reynolds numbers, the flow around the particle tends to separate producing vortices and wake forms, and the fluid inertia becomes more important. Nonetheless, the effect of flow separation is hardly noticeable until Reynolds number increases to about 20. The fluid motion at this point and beyond is quite different from Stokes flow, and the drag coefficient becomes higher than that predicted by Eq. (1). For very large Reynolds numbers nearly between 103 and 104, the viscous effect becomes so small to be neglected, and the drag coefficient approaches a constant ~ 0.4. The drag then increases slightly to ~ 0.5 at Re ≈ 5 × 104 and remains constant again. Up to here, the flow boundary condition is still considered laminar. The significant drop in drag coefficient at Re ≈ 2 × 105 represents the change in boundary layer from laminar to turbulent for smooth spheres. Beyond this, the drag coefficient starts decreasing until equal to ~ 0.09 at Re ≈ 5 × 105, and then it increases again to ~ 0.2 at Re ≈ 106. For Re N 106, experimental data of drag coefficient are not readily available; however, the flow may also become Reynolds number independent with a constant drag of ~ 0.2 that appears to be acceptable [18]. An analytical attempt to extend the range of approximation for the drag coefficient beyond Stokes flow is proposed by
Oseen [19] by including the inertia terms in the solution of Navier–Stokes CD ¼
24 3 1 þ Re Re 16
ð2Þ
Goldstein [20] considered more terms in the above relationship to obtain CD ¼
24 3 19 71 30; 179 1 þ Re Re2 þ Re3 Re4 þ N Re 16 1280 20; 480 34; 406; 400 ð3Þ
For both equations, reliable agreement with experimental data is found up to Re ≈ 2. Other attempts have also been presented in the literature, but they are still satisfactory only to a small range of Reynolds numbers (e.g. [10,21]). Although analytical attempts to the drag problem are available at small Reynolds values, only qualitative indications can be extended to larger ones. The reason is related to the complex flow pattern described earlier. Several researchers thus have established empirical relationships applicable under certain specified ranges. Table 1 presents suitable relationships in the literature with their Reynolds range of applicability. 3. Approach for asymptotic matching An approach is proposed here to derive an empirical relationship of drag coefficient of flow around a sphere valid for the entire range of Reynolds numbers reported in the literature, from stokes regime to the condition when turbulent boundary layer is dominant. For a relatively small segment, the correlation of drag coefficient with Particle Reynolds number can be considered as CD ¼ bRea
ð4Þ
220
J. Almedeij / Powder Technology 186 (2008) 218–223
The best-fitting constant b and exponent α for the segment can be determined by regression technique. The difficulty relies on matching asymptotically the many segments to develop an overall relationship representing the wide trend. For example, in Stokes range, the drag coefficient is proportional to Reynolds number with α = − 1 and b = 24 (Eq. (1)). As Reynolds number increases further towards ~ 104, α becomes closer to zero and b to 0.4. Within that range, a possible attempt to match asymptotically the upper and lower trends may be achieved by using the function φ = (24Re− 1)m + (0.4)m with the drag coefficient being in this case CD = (φ)1/m; where m is a constant determined by means of numerical optimization. This expression automatically satisfies the upper and lower trends at the very small and large Reynolds numbers, respectively; while in the middle, the trend needs to be tuned further. A more comprehensive function φ representing a series of segments of Eq. (4) can be expressed within that range by m m m u ¼ b0 Rea0 þ b1 Rea1 þ N þ bj Rea j ð5Þ
functions φ1, φ2,…, φj. The latter set may be combined into an overall relationship using a suitable composition expression 1=m ð7Þ CD ¼ f u1 ; u2 ; N ; uj The form of this expression should be chosen for CD to produce values equal to (φi)1/m if the computed Reynolds number is within the range applicable to φi. One comment is that, the choice of the points dividing the trend into φ1, φ2,…, φj should be considered with caution for the reason of every two subsequent functions not to intersect each other more than one time (at the dividing point) if extended to ∞ and − ∞, in order to avoid model collapse in the long run. Following this, the wide trend of drag coefficient can be divided into four functions φ1, φ2, φ3, and φ4 at the points Re = 104, 2 × 105, and 5 × 105; where φ1 (valid for the range of Re b 104) and φ3 (for 2 × 105 b Re b 5 × 105) are obtained by Expression (5), while φ2 (for 104 b Re b 2 × 105) and φ4 (for Re N 5 × 105) by Expression (6)
This model has α to be a real number and always monotonically increasing; however, this model cannot proceed further for very high flows when α starts decreasing. On the other hand, for α to be always monotonically decreasing, the function can be expressed by
10 10 10 u1 ¼ 24Re1 þ 21Re0:67 þ 4Re0:33 þð0:4Þ10 1 u2 ¼ 0:11 10 0:148Re þð0:5Þ10 10 u3 ¼ 1:57 108 Re1:625 1 u4 ¼ 10 17 6 10 Re2:63 þð0:2Þ10
1 m u¼ a0 m a1 m þ b1 Re þ N þ bj Reaj b0 Re
As an overall relationship, the following composition expression is adopted
a0 ba1 b N baj
ð6Þ
a0 Na1 N N Naj Given the above expressions, a relevant model of drag coefficient can be constructed. Expressions (5) and (6) have the potential to collect the adjacent segments of Eq. (4) and, therefore, to divide the wide trend of drag coefficient into subsequent
" CD ¼
1 ðu1 þ u2 Þ1 þðu3 Þ1
ð8Þ
#1=10 þ u4
ð9Þ
This expression renders CD equal to (φ1)1/10, (φ2)1/10, (φ3)1/10, or (φ4)1/10 when dealing with Reynolds values within the valid
Fig. 1. Drag coefficient for the wide range of Particle Reynolds numbers. Data shown in the figure obtained from Stokes regime by Eq. (1) and from experiments available in the literature [26,27].
J. Almedeij / Powder Technology 186 (2008) 218–223
range specified for the function at the beginning of this paragraph. Fig. 1 shows that the model has no systematic error compared to the experimental data available for Re b 106. The figure also shows that the model converges to Stokes solution at very small Reynolds numbers; while for Re N 106, the model renders a drag coefficient of 0.2. The accuracy of Eq. (9) can now be tested and compared to other relationships found in the literature. Although there are many methods to measure the model accuracy in an objective manner, the Mean Absolute Standard Error (MASE) is employed here n P
MASE ¼
sible to determine this variable in an indirect manner, it is convenient to have an explicit expression. An approach is proposed here to transform the drag coefficient relationship in the previous section into an explicit expression of particle settling velocity. Heywood [12] presented the force balance on a falling sphere in terms of the equations 4d 3 gD ¼ CD Re2 3v2
ð11Þ
3w3 Re ¼ 4gDv CD
ð12Þ
CDri
i¼1
ð10Þ
n
where
CDri
221
8 CDimeasured > > < C ¼ CDicalculated Dicalculated > > : CDimeasured
where Δ = (ρs − ρ) / ρ. The left hand sides of Eqs. (11) and (12) do not include the settling velocity and particle diameter, respectively. Those equations can be rewritten in dimensionless forms correspondingly as
if CDimeasured NCDicalculated if CDimeasured NCDicalculated
The above expression indicates that MASE ≥ 1. The closer the MASE value to one is, the better the accuracy of an equation will be, with MASE = 1 representing the condition of perfect agreement. Table 1 tests the accuracy of Eq. (9) and provides a comparison with the other relationships. As can be seen, Eq. (9) is superior not only in range, but also in accuracy with MASE = 1.02874. 4. Explicit relationship of settling velocity A very common practice is to use the results obtained for the drag coefficient to determine the settling velocity of a particle. Obviously, Eq. (9) is implicit in terms of settling velocity. While iterative techniques such as Newton–Raphson method are pos-
1=3
d⁎ ¼
3 CD Re2 4
w⁎ ¼
4 Re 1=3 3 CD
ð13Þ
ð14Þ
where d⁎ = d(gΔ/v2)1/3 and w⁎ = w(1/gΔv)1/3. Using the same experimental data in Fig. 1, the parameters d⁎ and w⁎ can plot Fig. 2 in a manner similar to that performed by Turton [28]. In Fig. 2, for a relatively small segment, the correlation of dimensionless settling velocity with particle diameter can be considered as w⁎ ¼ bd⁎a
ð15Þ
The constant b and exponent α for a specific segment in this figure can be determined by transforming the corresponding segment in Fig. 1 using Expressions (13) and (14). For example,
Fig. 2. Dimensionless settling velocity for the wide range of dimensionless particle diameters.
222
J. Almedeij / Powder Technology 186 (2008) 218–223
in Stokes range, substituting Eq. (1) into Expressions (13) and (14) and then manipulating them can yield w⁎ = 0.055 d⁎2. The remaining expressions for the segments in Fig. 2 can be determined similarly. Following the same approach in the previous section, the wide trend in Fig. 2 can then be divided into four functions of ψ1, ψ2, ψ3, and ψ4 as 1 w1 ¼ 10 10 10 10 2 1:256 þ 0:126d⁎ þ 0:518d⁎0:8 þ 1:826d⁎0:5 0:055d⁎ 10 10 w2 ¼ 2:834d⁎0:422 þ 1:633d⁎0:5 22 7 10 w3 ¼ 3 10 d⁎ 10 10 w4 ¼ 3393d⁎0:352 þ 2:582d⁎0:5 ð16Þ which can be combined, in turn, into an overall relationship by the composition expression 2 6 w⁎ ¼ 4
31=10 1 1 1 w1 1 þw2
þ w3
1
þw1 4
7 5
ð17Þ
The trend of Eq. (17) is presented in Fig. 2. It is worth mentioning that the performance of Eq. (17) is similar to that of Eq. (9) and results, thus, with MASE accuracy of ~ 1.02874. Eq. (17) can now be used to plot the dimensional settling velocity, w, as a function of particle diameter, d, and Kinematic viscosity, v (Fig. 3). It is interesting to note the effect of changing the boundary layer condition of a particle on the magnitude of settling velocity. The last trend in this figure with the lowest Kinematic viscosity of v = 10− 9 m2/s may be used to highlight more clearly this evaluation. As the boundary layer changes from laminar to turbulent, typically at Re ≈ 2 × 105, the settling velocity increases. This is seen at d = 1 mm with w value increased from 0.2 m/s to 0.485 m/s. For d N 1 mm, the difference
between the actual settling velocity calculated from the turbulent condition and the one obtained if the laminar boundary is assumed dominant decreases from ~ 59% to ~ 36%. The implication is that, a sufficiently large calculation error may be resulted if the relationship adopted for settling velocity does not account for the variation of boundary layer condition during very high flows, which is the case for many available models. 5. Conclusions An empirical relationship of drag coefficient of flow around a sphere was developed for the entire range of Particle Reynolds numbers reported in the literature from Stokes regime to the condition when turbulent boundary layer prevails. This relationship has been calibrated with experimental data for Reynolds numbers up to 10− 6. For higher Reynolds values, the relationship renders a drag coefficient of 0.2. The accuracy of the relationship has been tested and compared with other models in the literature. The drag coefficient relationship was also transformed into an explicit expression of particle settling velocity. It has been shown that for specific fluid and particle properties, if a settling velocity relationship disregards the variation of boundary layer condition during very high flows, a sufficiently large calculation error may be resulted. The expressions of drag coefficient and settling velocity were derived using an approach of asymptotic matching presented in this study. The approach, which is based on partitioning the wide trend of drag coefficient into smaller segments that can be combined into an overall relationship, employs regression techniques and thus warrants the best-fit accuracy results. This approach can possibly provide a means for modeling other phenomena with similar complex behaviors, including the drag coefficient of flow around non-spherical and deformable particles.
Fig. 3. Dimensional settling velocity as a function of particle diameter and Kinematic viscosity. The trends are estimated by Eq. (17) for Δ = 1.65.
J. Almedeij / Powder Technology 186 (2008) 218–223
List of symbols b, m constants drag coefficient CD d particle diameter d⁎ dimensionless particle diameter g gravitational acceleration n total number of data points Re Particle Reynolds number w settling velocity w⁎ dimensionless settling velocity
Greek symbols α constant v Kinematic viscosity ρ, ρs densities of water and particle φ, ψ functions
Acknowledgment This work was supported by Kuwait University, Research Grant No. EV06/05. References [1] J. Almedeij, M. Algharaib, Influence of sand production on pressure drawdown in horizontal wells: Theoretical evidence, J. Pet. Sci. Eng. 47 (2005) 137–145. [2] J.W. Delleur, New results and research needs on sediment movement in urban drainage, J. Water Resour. Plan. Manage., ASCE 127 (3) (2001) 186–193. [3] T. Hvitved-Jacobsen, J. Vollertsen, N. Tanaka, Wastewater quality changes during transport in sewers: An integrated aerobic and anaerobic model concept for carbon and sulfur microbial transformations, Water Sci. Technol. 38 (10) (1998) 257–264. [4] O. Mark, C. Appelgren, T. Larsen, Principles and approaches for numerical modeling of sediment transport in sewers, Water Sci. Technol. 31 (7) (1995) 107–115. [5] J. Almedeij, P. Diplas, Bed load sediment transport in ephemeral and perennial gravel bed streams, Eos 86 (44) (2005). [6] Z.X. Cao, Equilibrium near-bed concentration of suspended sediment, J. Hydraul. Eng., ASCE 125 (12) (1999) 1270–1278.
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[7] J.S. Bridge, S.J. Bennett, A model for the entrainment and transport of sediment grains of mixed sizes, shapes, and densities, Water Resour. Res. 28 (2) (1992) 337–363. [8] J. Duran, The physics of fine powders: Plugging and surface instabilities, C.R. Phys. 3 (2) (2002) 217–227. [9] J.G. Yates, Fundamentals of fluidized-bed processes, Butterworths, London, 1983. [10] S.J. Liao, An analytical approximation of the drag coefficient for the viscous flow past a sphere, Int. J. Non-linear Mech. 37 (2002) 1–18. [11] V.A. Vanoni, Sedimentation engineering: Theory, measurements, modeling, and practice, Manuals and Reports on Engineering Practice, 2nd edition, ASCE, vol. 54, 2006. [12] H. Heywood, J. Imp. Coll. Chem. Eng. Soc. 4 (1948) 17. [13] D.B. Simons, F. Senturk, Sediment transport technology: Water and sediment dynamics, Water Resources Publications, Colorado, 1992. [14] R.A. Khan, J.F. Richardson, The resistance to motion of a solid sphere in a fluid, Chem. Eng. Commun. 62 (1987) 135–150. [15] R. Turton, O. Levenspiel, A short note on the drag correlation of spheres, Powder Technol. 47 (1986) 83–86. [16] R.L.C. Flemmer, C.L. Banks, On the drag coefficient of a sphere, Powder Technol. 48 (1986) 217–221. [17] H. Rouse, Nomogram for the settling velocity of spheres, Report of the Committee on Sedimentation, National Research Council, Washington, 1937. [18] M.C. Potter, D.C. Wiggert, Mechanics of fluids, 3rd edition, 2002 Brooks/ Cole. [19] C. Oseen, Hydrodynamik, Chapter 10, Akademische Verlagsgesellschaft, Leipzig, 1927. [20] S. Goldstein, The steady flow of viscous fluid past a fixed spherical obstacle at small Reynolds numbers, Proc. R. Soc. Lond. 123A (1929). [21] I. Proudman, J. Pearson, Expansions of small Reynolds numbers for the flow past a sphere and a circular cylinder, J. Fluid Mech. 2 (1957) 237–262. [22] L. Schiller, A.Z. Nauman, Über die grundlegenden berechungen bei der schwerkraftaufbereitung, Ver. Deut. Ing. 77 (1933) 318–320. [23] L.B. Torobin, W.H. Gauvin, Fundamental aspects of solidnngas flow, Can. J. Chem. Eng. 37 (4) (1959) 129–141. [24] R.P. Hesketh, A.W. Etchells, T.W. Russell, Bubble breakage in pipeline flow, Chem. Eng. Sci. 46 (1) (1991) 1–9. [25] K. Ceylan, A. Altunbas, C. Kelbaliyev, A new model for estimation of drag force in the flow of Newtonian fluids around rigid or deformable particles, Powder Technol. 119 (2001) 250–256. [26] F.W. Roos, W.W. Willmarth, Some experimental results on sphere and disk drag, AIAA J. 9 (2) (1971) 285–291. [27] V.M. Voloshuk, J.S. Sedunow, The processes of coagulation in dispersed systems, Nauka, Moscow, 1971. [28] R. Turton, N.N. Clark, An explicit relationship to predict spherical particle terminal velocity, Powder Technol. 53 (1987) 127–129.
Powder Technology 186 (2008) 218 – 223 www.elsevier.com/locate/powtec
Drag coefficient of flow around a sphere: Matching asymptotically the wide trend Jaber Almedeij ⁎ Civil Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received 3 May 2007; received in revised form 7 December 2007; accepted 10 December 2007 Available online 21 February 2008
Abstract An empirical relationship of drag coefficient of flow around a sphere is developed for the entire range of Particle Reynolds numbers reported in the literature from Stokes regime to the condition when turbulent boundary layer prevails. The relationship is obtained using an approach to match asymptotically the wide trend of drag coefficient. The matching approach, which relies on dividing the wide trend into smaller segments that can be combined into an overall relationship, employs regression techniques and thus warrants the best-fit accuracy results. The relationship is calibrated with experimental data available in the literature covering the entire range for Reynolds numbers up to ~ 106. For Reynolds values greater than 106, the relationship renders a drag coefficient of 0.2. The performance of the relationship is tested and compared with other suitable models found in the literature. This relationship is also transformed into an explicit expression for settling velocity calculations. © 2008 Elsevier B.V. All rights reserved. Keywords: Regression; Reynolds number; Settling velocity; Turbulent boundary; Viscous flow
1. Introduction For the case of flow around a sphere, certain hydraulic analyses require determining the drag coefficient as a function of Particle Reynolds number. This is, for example, worthy to estimate the particle settling velocity, which is a parameter required for the diverse implications of sediment transport and deposition in pipelines (e.g. [1–4]) and alluvial channels (e.g. [5–7]), and of chemical and powder processes (e.g. [8,9]). However, most research efforts report existing difficulties to model theoretically the relationship of drag coefficient. One problem is that the drag coefficient cannot be expressed in an analytical form for a wide range of Particle Reynolds numbers, because the flow condition during the process is highly complicated [10]. This relationship though can be provided in charts and tables [12], determined experimentally by observing the settling velocities in still fluids or by measuring the drag of spheres in wind and tunnels [11]. Owing to the high advances in the development of computer and software applications, the numerical ⁎ Tel.: +965 4987603; fax: +965 4817524. E-mail address: [email protected]. 0032-5910/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.12.006
data in charts and tables representing this relationship would not be practical for the fast computational schemes. Rather, a numerical expression would be necessary. Attempts have thus been made to express the relationship empirically in order to extend the range of prediction. Until now, most of the empirical expressions are not satisfactory [13,14]. Only few empirical attempts, although they are valid for restricted ranges of Reynolds numbers, present acceptable drag coefficient results (e.g. [15,16]). The aim of the study is to derive an empirical relationship of drag coefficient of flow around a sphere valid for all Particle Reynolds numbers. Initially, the relationship is derived by using an approach to match asymptotically the wide trend of drag coefficient. Then, the accuracy of the relationship is tested and compared with other suitable models in the literature. The relationship is then transformed into an explicit expression of settling velocity as a function of particle diameter. 2. Theory and proposed relationships A graphical presentation of the experimental data of drag cofficient, CD, as a function of Particle Reynolds number, Re = wd/v, found in the vast literature is that given by Rouse [17].
J. Almedeij / Powder Technology 186 (2008) 218–223
219
Table 1 Summary of some empirical relationships, range of applicability, and accuracy Author Schiller et al. [22] Torobin et al. [23] Flemmer and Banks [16]
Turton and Levenspiel [15] Hesketh et al. [24]
Empirical relationship 0:687 24 CD ¼ Re 1 þ 0:15Re 0:63 24 þ 0:0026Re1:38 CD ¼ Re 1 þ 0:197Re E 24 CD ¼ Re 10 where E ¼ 0:261Re0:369 0:105Re0:431 1þð0:124 log ReÞ2 24 0:413 CD ¼ Re 1 þ 0:173Re0:657 þ 1þ16300Re 1:09 h i2 CD ¼ 0:352 þ ð0:124 þ 24=ReÞ1=2
5
a b
MASE a
2 b Re b 800 1 b Re b 100 Re b 3 × 105
1.03102 1.14055 1.03538 b
Re b 2 × 105
1.03391 b
1=4 0.1 b Re b 104 CD ¼ 1 0:5eð0:182Þ þ 10:11Re2=3 eð0:952Re Þ 0:125104 ReÞ 4 4=3 ð1:3Re1=2 Þ ð 0:03859Re e þ 0:037 10 Re e
0:116 1010 Re2 eð0:44410
Ceylan et al. [25] Almedeij
Range of applicability
1.09531
ReÞ
0.1 b Re b 106 Re b 106
Eq. (9)
1.08306 1.02874 b
Obtained using Eq. (10) and for the range of experimental data within which the model is valid. Calculated from experimental data starting from Re = 0.0001.
As the particle size or flow velocity increases for a given Kinematic viscosity, so does the Reynolds number, and then the character of flow changes. For very small Reynolds numbers, Stokes proposed an analytical solution of drag coefficient by solving the general differential equation of Navier–Stokes CD ¼
24 Re
ð1Þ
The Stokes solution neglects the effects of inertia and is acceptable roughly for Re b 0.4, when the laminar boundary layer is not separated from the particle. This solution reflects entirely the viscous effect of flow. For larger Reynolds numbers, the flow around the particle tends to separate producing vortices and wake forms, and the fluid inertia becomes more important. Nonetheless, the effect of flow separation is hardly noticeable until Reynolds number increases to about 20. The fluid motion at this point and beyond is quite different from Stokes flow, and the drag coefficient becomes higher than that predicted by Eq. (1). For very large Reynolds numbers nearly between 103 and 104, the viscous effect becomes so small to be neglected, and the drag coefficient approaches a constant ~ 0.4. The drag then increases slightly to ~ 0.5 at Re ≈ 5 × 104 and remains constant again. Up to here, the flow boundary condition is still considered laminar. The significant drop in drag coefficient at Re ≈ 2 × 105 represents the change in boundary layer from laminar to turbulent for smooth spheres. Beyond this, the drag coefficient starts decreasing until equal to ~ 0.09 at Re ≈ 5 × 105, and then it increases again to ~ 0.2 at Re ≈ 106. For Re N 106, experimental data of drag coefficient are not readily available; however, the flow may also become Reynolds number independent with a constant drag of ~ 0.2 that appears to be acceptable [18]. An analytical attempt to extend the range of approximation for the drag coefficient beyond Stokes flow is proposed by
Oseen [19] by including the inertia terms in the solution of Navier–Stokes CD ¼
24 3 1 þ Re Re 16
ð2Þ
Goldstein [20] considered more terms in the above relationship to obtain CD ¼
24 3 19 71 30; 179 1 þ Re Re2 þ Re3 Re4 þ N Re 16 1280 20; 480 34; 406; 400 ð3Þ
For both equations, reliable agreement with experimental data is found up to Re ≈ 2. Other attempts have also been presented in the literature, but they are still satisfactory only to a small range of Reynolds numbers (e.g. [10,21]). Although analytical attempts to the drag problem are available at small Reynolds values, only qualitative indications can be extended to larger ones. The reason is related to the complex flow pattern described earlier. Several researchers thus have established empirical relationships applicable under certain specified ranges. Table 1 presents suitable relationships in the literature with their Reynolds range of applicability. 3. Approach for asymptotic matching An approach is proposed here to derive an empirical relationship of drag coefficient of flow around a sphere valid for the entire range of Reynolds numbers reported in the literature, from stokes regime to the condition when turbulent boundary layer is dominant. For a relatively small segment, the correlation of drag coefficient with Particle Reynolds number can be considered as CD ¼ bRea
ð4Þ
220
J. Almedeij / Powder Technology 186 (2008) 218–223
The best-fitting constant b and exponent α for the segment can be determined by regression technique. The difficulty relies on matching asymptotically the many segments to develop an overall relationship representing the wide trend. For example, in Stokes range, the drag coefficient is proportional to Reynolds number with α = − 1 and b = 24 (Eq. (1)). As Reynolds number increases further towards ~ 104, α becomes closer to zero and b to 0.4. Within that range, a possible attempt to match asymptotically the upper and lower trends may be achieved by using the function φ = (24Re− 1)m + (0.4)m with the drag coefficient being in this case CD = (φ)1/m; where m is a constant determined by means of numerical optimization. This expression automatically satisfies the upper and lower trends at the very small and large Reynolds numbers, respectively; while in the middle, the trend needs to be tuned further. A more comprehensive function φ representing a series of segments of Eq. (4) can be expressed within that range by m m m u ¼ b0 Rea0 þ b1 Rea1 þ N þ bj Rea j ð5Þ
functions φ1, φ2,…, φj. The latter set may be combined into an overall relationship using a suitable composition expression 1=m ð7Þ CD ¼ f u1 ; u2 ; N ; uj The form of this expression should be chosen for CD to produce values equal to (φi)1/m if the computed Reynolds number is within the range applicable to φi. One comment is that, the choice of the points dividing the trend into φ1, φ2,…, φj should be considered with caution for the reason of every two subsequent functions not to intersect each other more than one time (at the dividing point) if extended to ∞ and − ∞, in order to avoid model collapse in the long run. Following this, the wide trend of drag coefficient can be divided into four functions φ1, φ2, φ3, and φ4 at the points Re = 104, 2 × 105, and 5 × 105; where φ1 (valid for the range of Re b 104) and φ3 (for 2 × 105 b Re b 5 × 105) are obtained by Expression (5), while φ2 (for 104 b Re b 2 × 105) and φ4 (for Re N 5 × 105) by Expression (6)
This model has α to be a real number and always monotonically increasing; however, this model cannot proceed further for very high flows when α starts decreasing. On the other hand, for α to be always monotonically decreasing, the function can be expressed by
10 10 10 u1 ¼ 24Re1 þ 21Re0:67 þ 4Re0:33 þð0:4Þ10 1 u2 ¼ 0:11 10 0:148Re þð0:5Þ10 10 u3 ¼ 1:57 108 Re1:625 1 u4 ¼ 10 17 6 10 Re2:63 þð0:2Þ10
1 m u¼ a0 m a1 m þ b1 Re þ N þ bj Reaj b0 Re
As an overall relationship, the following composition expression is adopted
a0 ba1 b N baj
ð6Þ
a0 Na1 N N Naj Given the above expressions, a relevant model of drag coefficient can be constructed. Expressions (5) and (6) have the potential to collect the adjacent segments of Eq. (4) and, therefore, to divide the wide trend of drag coefficient into subsequent
" CD ¼
1 ðu1 þ u2 Þ1 þðu3 Þ1
ð8Þ
#1=10 þ u4
ð9Þ
This expression renders CD equal to (φ1)1/10, (φ2)1/10, (φ3)1/10, or (φ4)1/10 when dealing with Reynolds values within the valid
Fig. 1. Drag coefficient for the wide range of Particle Reynolds numbers. Data shown in the figure obtained from Stokes regime by Eq. (1) and from experiments available in the literature [26,27].
J. Almedeij / Powder Technology 186 (2008) 218–223
range specified for the function at the beginning of this paragraph. Fig. 1 shows that the model has no systematic error compared to the experimental data available for Re b 106. The figure also shows that the model converges to Stokes solution at very small Reynolds numbers; while for Re N 106, the model renders a drag coefficient of 0.2. The accuracy of Eq. (9) can now be tested and compared to other relationships found in the literature. Although there are many methods to measure the model accuracy in an objective manner, the Mean Absolute Standard Error (MASE) is employed here n P
MASE ¼
sible to determine this variable in an indirect manner, it is convenient to have an explicit expression. An approach is proposed here to transform the drag coefficient relationship in the previous section into an explicit expression of particle settling velocity. Heywood [12] presented the force balance on a falling sphere in terms of the equations 4d 3 gD ¼ CD Re2 3v2
ð11Þ
3w3 Re ¼ 4gDv CD
ð12Þ
CDri
i¼1
ð10Þ
n
where
CDri
221
8 CDimeasured > > < C ¼ CDicalculated Dicalculated > > : CDimeasured
where Δ = (ρs − ρ) / ρ. The left hand sides of Eqs. (11) and (12) do not include the settling velocity and particle diameter, respectively. Those equations can be rewritten in dimensionless forms correspondingly as
if CDimeasured NCDicalculated if CDimeasured NCDicalculated
The above expression indicates that MASE ≥ 1. The closer the MASE value to one is, the better the accuracy of an equation will be, with MASE = 1 representing the condition of perfect agreement. Table 1 tests the accuracy of Eq. (9) and provides a comparison with the other relationships. As can be seen, Eq. (9) is superior not only in range, but also in accuracy with MASE = 1.02874. 4. Explicit relationship of settling velocity A very common practice is to use the results obtained for the drag coefficient to determine the settling velocity of a particle. Obviously, Eq. (9) is implicit in terms of settling velocity. While iterative techniques such as Newton–Raphson method are pos-
1=3
d⁎ ¼
3 CD Re2 4
w⁎ ¼
4 Re 1=3 3 CD
ð13Þ
ð14Þ
where d⁎ = d(gΔ/v2)1/3 and w⁎ = w(1/gΔv)1/3. Using the same experimental data in Fig. 1, the parameters d⁎ and w⁎ can plot Fig. 2 in a manner similar to that performed by Turton [28]. In Fig. 2, for a relatively small segment, the correlation of dimensionless settling velocity with particle diameter can be considered as w⁎ ¼ bd⁎a
ð15Þ
The constant b and exponent α for a specific segment in this figure can be determined by transforming the corresponding segment in Fig. 1 using Expressions (13) and (14). For example,
Fig. 2. Dimensionless settling velocity for the wide range of dimensionless particle diameters.
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J. Almedeij / Powder Technology 186 (2008) 218–223
in Stokes range, substituting Eq. (1) into Expressions (13) and (14) and then manipulating them can yield w⁎ = 0.055 d⁎2. The remaining expressions for the segments in Fig. 2 can be determined similarly. Following the same approach in the previous section, the wide trend in Fig. 2 can then be divided into four functions of ψ1, ψ2, ψ3, and ψ4 as 1 w1 ¼ 10 10 10 10 2 1:256 þ 0:126d⁎ þ 0:518d⁎0:8 þ 1:826d⁎0:5 0:055d⁎ 10 10 w2 ¼ 2:834d⁎0:422 þ 1:633d⁎0:5 22 7 10 w3 ¼ 3 10 d⁎ 10 10 w4 ¼ 3393d⁎0:352 þ 2:582d⁎0:5 ð16Þ which can be combined, in turn, into an overall relationship by the composition expression 2 6 w⁎ ¼ 4
31=10 1 1 1 w1 1 þw2
þ w3
1
þw1 4
7 5
ð17Þ
The trend of Eq. (17) is presented in Fig. 2. It is worth mentioning that the performance of Eq. (17) is similar to that of Eq. (9) and results, thus, with MASE accuracy of ~ 1.02874. Eq. (17) can now be used to plot the dimensional settling velocity, w, as a function of particle diameter, d, and Kinematic viscosity, v (Fig. 3). It is interesting to note the effect of changing the boundary layer condition of a particle on the magnitude of settling velocity. The last trend in this figure with the lowest Kinematic viscosity of v = 10− 9 m2/s may be used to highlight more clearly this evaluation. As the boundary layer changes from laminar to turbulent, typically at Re ≈ 2 × 105, the settling velocity increases. This is seen at d = 1 mm with w value increased from 0.2 m/s to 0.485 m/s. For d N 1 mm, the difference
between the actual settling velocity calculated from the turbulent condition and the one obtained if the laminar boundary is assumed dominant decreases from ~ 59% to ~ 36%. The implication is that, a sufficiently large calculation error may be resulted if the relationship adopted for settling velocity does not account for the variation of boundary layer condition during very high flows, which is the case for many available models. 5. Conclusions An empirical relationship of drag coefficient of flow around a sphere was developed for the entire range of Particle Reynolds numbers reported in the literature from Stokes regime to the condition when turbulent boundary layer prevails. This relationship has been calibrated with experimental data for Reynolds numbers up to 10− 6. For higher Reynolds values, the relationship renders a drag coefficient of 0.2. The accuracy of the relationship has been tested and compared with other models in the literature. The drag coefficient relationship was also transformed into an explicit expression of particle settling velocity. It has been shown that for specific fluid and particle properties, if a settling velocity relationship disregards the variation of boundary layer condition during very high flows, a sufficiently large calculation error may be resulted. The expressions of drag coefficient and settling velocity were derived using an approach of asymptotic matching presented in this study. The approach, which is based on partitioning the wide trend of drag coefficient into smaller segments that can be combined into an overall relationship, employs regression techniques and thus warrants the best-fit accuracy results. This approach can possibly provide a means for modeling other phenomena with similar complex behaviors, including the drag coefficient of flow around non-spherical and deformable particles.
Fig. 3. Dimensional settling velocity as a function of particle diameter and Kinematic viscosity. The trends are estimated by Eq. (17) for Δ = 1.65.
J. Almedeij / Powder Technology 186 (2008) 218–223
List of symbols b, m constants drag coefficient CD d particle diameter d⁎ dimensionless particle diameter g gravitational acceleration n total number of data points Re Particle Reynolds number w settling velocity w⁎ dimensionless settling velocity
Greek symbols α constant v Kinematic viscosity ρ, ρs densities of water and particle φ, ψ functions
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