Cheng, N. S. (1997). "A simplified settling velocity formula for sediment particle." Journal of Hydraulic Engineering, ASCE, 123(2),

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1 THIS PAPER IS CITED AS Cheng, N. S. (1997). "A simplifie settling velocity formula for seiment particle." Journal of Hyraulic Engineering, ASCE, 13(), A SIMPLIFIED SETTLING VELOCITY FORMULA FOR SEDIMENT PARTICLE By Nian-Sheng Cheng 1 ABSTRACT: A new an simplifie formula for preicting the settling velocity of natural seiment particles is evelope. The formula proposes an explicit relationship between the particle ynols number an a imensionless particle parameter. It is applicable to a wie range of ynols numbers from the Stokes flow to the turbulent regime. The propose formula has the highest egree of preiction accuracy when compare with other publishe formulas. It also agrees well with the wiely use iagrams an tables propose by the U. S. Inter-Agency Committee (1957). INTRODUCTION A pre-requisite to certain quantitative analysis in seiment transport is a knowlege of the settling velocity of seiment particles. Many attempts have been mae for its preiction but most of the relevant researches apply only to spherical particles. Basically, there are two types of preiction methos for settling velocity of either spherical or non-spherical particles. One is the analytical solution of Stokes which is applicable only for particle ynols number, = w/v 1, where w = settling velocity of a particle, = particle iameter an v = kinematic viscosity. The other inclues tabulate ata an iagrams consisting of families of curves base on the experimental ata, e.g. Schiller an Naumann (1933) an U. S. Inter-Agency Committee (1957). They are suite to a wie range of ynols numbers but inconvenient to use in practice. The objective of this note is to erive a simple expression for the etermination of the settling velocity of natural seiment particles over a wie range. 1 School of Civil an Structural Engrg., Nanyang Technological University, Nanyang Avenue, Singapore, Key wors: settling velocity, seiment particle, rag coefficient, ynols number 1

2 DRAG COEFFICIENT FOR SETTLING OF INDIVIDUAL PARTICLE In 1851, Stokes obtaine the solution for the rag resistance of flow past a sphere by expressing the simplifie Navier-Stokes equation together with the continuity equation in polar co-orinates. Using his solution, the following expression for the settling velocity of spherical particles can be erive as, g (1) 18 ν where, = (ρ S - ρ)/ρ, ρ = ensity of flui, ρ S = ensity of particles, g = gravitational acceleration. Unfortunately, (1) is only vali for 1. Generally, by equating the effective weight force to the Newtonian expression of rag resistance, i.e. π 3 π ρw ( ρs ρ) g = CD () 6 4 the rag coefficient C D can be expresse as C D = 4 g (3) 3 w By substituting the Stokes solution in (1) into (3), C D may be relate to the ynols number: A C D = (4) where A = a constant, which is epenent on the shape factor of the particle. For Stokes solution, A = 4 for spherical particles. The effect of particle shape on the rag coefficient varies, being small at low an more appreciable at high (Schulz, et al., 1954). Usually the shape factor of seiment particle is less than unity an for natural san particles, the shape factor is about 0.7. Table 1 shows the value of A to be about 3 base on the work of various investigators. Uner the conition of high ynols numbers, say, 10 3 ~ 10 5, the rag coefficient of spheres has an average value of about 0.4. For natural seiment particles, C D lies between 1.0 ~ 1. as shown in Table 1. As the Stokes-type equation is restricte to 1, efforts have been mae to evelop a metho for extening (4) to a much wier range of ynols numbers. Some quasi-theoretical formulas or empirical correlations for evaluating the settling velocity of iniviual particle can be foun in the literature, e.g., Oseen (197), Sha (1956), Zanke (1977) an Raukivi (1990). In the light of all these stuies, the following relation between C D an is assume for natural seiment particles: 1 1 A C = [( n n n D ) +B ] (5) where A an B are constants, an n = an exponent. Eq. (5) automatically satisfies the two extreme conitions at low an high ynols numbers, that is, C D is inversely proportional to at low ynols numbers an becomes a constant at high ynols numbers. Accoring to Table 1, A can be taken as 3, as most researchers i, an B = 1, being the lowest limit of the rag coefficient for seiment particles. As the relationship between C D an at the extreme ynols numbers is unaffecte appreciably by n in (5), the latter may be estimate by fitting (5) with the experimental ata in the intermeiate ynols number range, i.e., 1 < < Base on the experimental ata of Concharov (196: see Iba-zae, 199), Zegzha (1934), Arkhangel skii (1935) an Sarkisyan (1958) for quartz san particles, the average n-value was foun to be 1.5. Therefore, with the foregoing values propose for A an B, (5) can be rewritten as C D = [( ) + 1] (6)

3 Eq. (6) is a general relationship between the rag coefficient an particle ynols number for natural seiment particles. Using a imensionless particle parameter * efine as g * = ( 1 3 ) (7) ν together with (3), we have C = 4 3 * D (8) 3 Substituting (8) into (6) yiels w ν = ( +. ) * 5 (9) Eq. (9) can be use to evaluate the settling velocity of natural san particles explicitly. COMPARISON WITH PREVIOUS STUDIES There are numerous settling velocity formulas evelope by ifferent investigators for spherical an non-spherical particles. The settling velocity formulas for seiment particles use for comparison in this note are as follows: 1. Sha (1954) g for < 0.01 cm (10a) 4 ν w= 114. g for > 0. cm (10b) (log ) + (log * ) = 39 * for = 0.01 ~ 0. cm (10c). Concharov (196: see Iba-zae, 199) g 4 ν for < cm (11a) w= g for > 0.15 cm (11b) T w= ( 1) 6 for = ~ 0.15 cm (11c) In (11c), the temperature T is in o C, in cm an w in cm/s. 3. Zhang (1989) w = ν ν ( ) g (1) 4. Van Rijn (1989) g 18 ν for < 0.01 cm (13a) w= 11. g for > 0.1 cm (13b) ν w = 10 ( * 1 ) (Zanke, 1977) for = 0.01 ~ 0.1 cm (13c) 5. Zhu an Cheng (1993) w α α α α ν cos cos ( cos 3. 6 sin ) * 3 9cos α sin α (14) where α = 0 for * 1 an α = π/(+.5(log * ) -3 ) for * > 1. To test the accuracy of (9) an the others from (10) to (14), three ata sets for san particles were use. The first is the relations an table in Raukivi (1990), on the average 3

4 settling velocity of quartz san particles in water at 0 o C. Table shows the 13 sets ata points which are compute using the metho outline in Raukivi (1990). The secon is the experimental ata of Zegzha (1934), Arkhangel skii (1935) an Sarkisyan (1958), which were compile in the orer of ecreasing particle iameter by Zhu an Cheng (1993), as shown in Table 3. The last is the tabulate ata given by the U. S. Inter-Agency Committee (1957) (also see: Raukivi, 1990) for settling velocity of natural seiment particles with a shape factor of 0.7, an specific gravity ranging from.0 to 4.3. The basic parameter use for the etermination of accuracy of a formula is the average value of the relative error efine as calculate given error = 100 (15) given Table presents the comparison of the calculate settling velocity using (9) together with those using the other five methos, with the average values reprouce from Raukivi (1990). It can be seen that (9) has the smallest relative error when compare with the other formulas. The comparison given in Table 3 is between the various compute results an the experimental ata of Zegzha (1934), Arkhangel skii (1935) an Sarkisyan (1958). It shows that the average relative error of (9) is 6.1%, which is very close to the 5.8% achieve by Zhu an Cheng s (1993) formula an the egree of accuracy is better than all the other formulas. The present formula is also simpler to use than that propose earlier by Zhu an Cheng (1993). Fig. 1 isplays the relationship of an * erive from (9) an it can be seen that the compute ata also agree very well with the tabulate ones given by the U. S. Inter-Agency Committee (1957). CONCLUSIONS An explicit an simple formula was evelope for evaluating the settling velocity of iniviual natural seiment particles. The formula is applicable to the ifferent regimes ranging from the Stokes flow to the high ynols number. Comparison with publishe experimental ata shows that the propose formula has a high egree of preiction accuracy. ACKNOWLEDGEMENTS The author is thankful to Dr. Siow-Yong Lim an Dr. Yee-Meng Chiew, School of Civil an Structural Engineering, Nanyang Technological University, an the anonymous referees for their reviews an useful comments. APPENDIX I. REFERENCES 1. Arkhangel skii, B. V. (1935). Experimental stuy of accuracy of hyraulic coarseness scale of particles. Izv. NIIG, No. 15, Moscow (in Russian).. Iba-zae, Y. A. (199). Movement of seiment in open channels. Translate by Ghosh, S. P. Russian Translations Series, Vol. 49. A. A. Balkema/Rotteram. 3. Oseen, C. W. (197). Neuere Methoen un Ergebnisse in er Hyroynamik. Akaemische Verlagsgesellschaft, Leipzig. 4. Raukivi, A. J. (1990). Loose bounary hyraulics. 3r eition, Pergamon Press. 5. Sarkisyan, A. A. (1958). Deposition of seiment in a turbulent stream. Iz. AN SSSR, Moscow (in Russian). 6. Schiller, L., an Naumann, A. (1933). Uber ie grunlegenen Berechnungen bei er Schwekraftaubereitung. Zeitschrift es Vereines Deutscher Ingenieure, 77(1), Schulz, S. F., Wile, R. H., an Albertson, M. L. (1954). Influence of shape on the fall velocity of seimentary particles. M. R. D. Seiment Series, No. 5, Missouri River Div., U. S. Corps of Engrs. 8. Sha, Y. Q. (1956). Basic principles of seiment transport. Journal of Seiment search. 1(): (in Chinese). 9. U. S. Inter-Agency Committee (1957). Some funamentals of particle size analysis. A 4

5 stuy of methos use in measurement an analysis of seiment loas in streams. Subcommittee on Seimentation, U. S. Inter-Agency Committee on Water sources, port No. 1, St. Anthony Falls Hyr. Lab., Minneapolis, Minn. 10. Van Rijn, L. C. (1989). Hanbook: Seiment transport by currents an waves. port H461, Delft Hyraulics, Netherlans. 11. Zanke, U. (1977). Berechnung er Sinkgeschwinigkeiten von Seimenten. Mitt. Des Franzius-Instituts fur Wasserbau, Heft 46, Seite 43, Technical University, Hannover, Deutshlan. 1. Zegzha, A. P. (1934). Settlement of san gravel particles in still water. Izv. NIIG, No.1 Moscow (in Russian). 13. Zhang, R. J. (1989). Seiment ynamics in rivers. Water sources Press. (in Chinese). 14. Zhu, L. J., an Cheng, N. S. (1993). Settlement of seiment particles. search port, Dept. of River an Harbour Engrg., Nanjing Hyr. s. Institute, China. (in Chinese). APPENDIX II. NOTATIONS The following symbols are use in this paper: A,B = constants; C D = rag coefficient; = iameter of a particle; * = imensionless particle parameter; g = gravitational acceleration; n = exponent; = w/ν = particle ynols number; T = temperature; w = settling velocity of a particle; α = parameter; = (ρ S - ρ)/ρ; ν = kinematic viscosity of flui; ρ = ensity of flui; an ρ S = ensity of seiment particles. 5

6 U. S. Inter-Agency Committee (1957) Eq. (9) Fig. 1 Comparison between (9) an the U. S. Inter-Agency Committee Data, 1957 * 6

7 Table 1 Drag Coefficient of Seiment Particles at Extreme ynols Numbers Author C D (low ) C D (high ) Sha (1956) 3/ 1.0 Concharov (196) 3/ 1. Zhang (1989) 34/ 1. Van Rijn (1989) 4/ 1.1 Raukivi (1990) 3/ 1. Zhu an Cheng (1993) 3/ 1. 7