TECHNICAL PAPERS
Feb 14, 2003

Sphere Drag and Settling Velocity Revisited

Publication: Journal of Environmental Engineering
Volume 129, Issue 3

Abstract

Sphere drag data from throughout the twentieth century are available in tabular form. However, much of the data arose from experiments in small diameter cylindrical vessels, where the results might have been influenced by the wall effect. Wall effect corrections developed by others were applied to 178 of the 480 data points collected. This corrected data set is believed to be free of the influence of wall effects. Existing drag and settling velocity correlations were compared to this data set. In addition, new correlations of the same forms were developed using the corrected data. Two new correlations of sphere terminal velocity are proposed, one applicable for all Reynolds numbers less than 2×105, and the other designed to predict settling velocities with exceptional accuracy for terminal Reynolds numbers less than 4,000, a region that contains almost all applications of interest in environmental engineering. The trial and error solution for settling velocity using the Fair and Geyer equation for drag should be retired in favor of the direct calculation available from these new correlations.

Get full access to this article

View all available purchase options and get full access to this article.

References

Achenbach, E.(1972). “Experiments on the flow past spheres at very high Reynolds numbers.” J. Fluid Mech., 54, 565–575.
Allen, H. S.(1900). “The motion of a sphere in a viscous fluid.” Philos. Mag., 50, 519–534.
American Water Works Association. (1999). Water quality and treatment, 5th Ed., McGraw-Hill, New York.
Arnold, H. D.(1911). “Limitations imposed by slip and inertia terms upon stokes’s law for the motion of spheres through liquids.” Philos. Mag., 22, 755–775.
Bailey, A. B.(1974). “Sphere drag coefficient for subsonic speeds in continuum and free-molecule flows.” J. Fluid Mech., 65(2), 401–410.
Bailey, A. B., and Hiatt, J. (1971). “Free-flight measurements of sphere drag at subsonic, transonic, supersonic, and hypersonic speeds for continuum, transition, and near-free-molecular flow conditions.” Rep. No. AEDC-TR-70-291, Arnold Engineering Development Center, Arnold Air Force Station, Tenn.
Bailey, A. B., and Hiatt, J.(1972). “Sphere drag coefficients for a broad range of mach and Reynolds numbers.” AIAA J., 10, 1436–1440.
Beard, K. V., and Pruppacher, H. R.(1969). “A determination of the terminal velocity and drag of small water drops by means of a wind tunnel.” J. Atmos. Sci., 26, 1066–1072.
Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles, drops, and particles, Academic, New York.
Davies, C. N.(1945). “Definitive equations for the fluid resistance of spheres.” Proc. Phys. Soc., London, 57, 259–270.
Dennis, S. C. R., and Walker, J. D. A.(1971). “Calculation of the steady flow past a sphere at low and moderate Reynolds numbers.” J. Fluid Mech., 48, 771–789.
Droste, R. L. (1997). Theory and practice of water and wastewater treatment, Wiley, New York.
Fair, G., and Geyer, J. (1954). Water supply and waste water disposal, Wiley, New York.
Fidleris, V., and Whitmore, R. L.(1961). “Experimental determination of the wall effect for spheres falling axially in cylindrical vessels.” Br. J. Appl. Phys., 12, 490–494.
Flemmer, R. L. C., and Banks, C. L.(1986). “On the drag coefficient of a sphere.” Powder Technol., 48(3), 217–221.
Goin, K. L., and Lawrence, W. R.(1968). “Subsonic drag of spheres at Reynolds numbers from 200 to 10,000.” AIAA J., 6, 961–962.
Gunn, R., and Kinzer, G. D.(1949). “The terminal velocity of fall for water droplets in stagnant air.” J. Meteorol., 6, 243–250.
Haider, A., and Levenspiel, O.(1989). “Drag coefficient and terminal velocity of spherical and nonspherical particles.” Powder Technol., 58, 63–70.
Hartman, M., Trnka, O., and Svoboda, K.(1994). “Free settling of nonspherical particles.” Ind. Eng. Chem. Res., 33, 1979–1983.
Karamanev, D. G.(1996). “Equations for calculation of the terminal velocity and drag coefficient of solid spheres and gas bubbles.” Chem. Eng. Commun., 147, 75–84.
Khan, A. R., and Richardson, J. F.(1987). “The resistance to motion of a solid sphere in a fluid.” Chem. Eng. Commun., 62, 135–150.
Liebster, H.(1927). “Uber den widerstand von kugeln.” Annalen der Physik (Leipzig), 82, 541–562.
Lunnon, R. G.(1928). “Fluid resistance to moving spheres.” Proc. R. Soc. London, Ser. A, 118, 680–694.
Maxworthy, T.(1965). “Accurate measurements of sphere drag at low reynolds numbers.” J. Fluid Mech., 23, 369–372.
McGauhey, P. H.(1956). “Theory of sedimentation.” J. Am. Water Works Assoc., 48, 437–448.
Metcalf and Eddy, Inc. (1991). Wastewater engineering: Treatment, disposal, and reuse, 3rd Ed., McGraw-Hill, New York.
Moller, W.(1938). “Experimentelle untersuchungen zur hydrodynamik der kugel.” Phys. Z., 39, 57–80.
Montgomery Consulting Engineers, Inc. (1985). Water treatment principles and design, Wiley, New York.
Pettyjohn, E. S., and Christiansen, E. B.(1948). “Effect of particle shape on free-settling rates of isometric particles.” Chem. Eng. Prog., 44, 157–172.
Pruppacher, H. R., and Steinberger, E. H.(1968). “Experimental determination of the drag on a sphere at low Reynolds numbers.” J. Appl. Phys., 39(9), 4129–4132.
Reynolds, T. D., and Richards, P. A. (1996). Unit operations and processes in environmental engineering, 2nd Ed., PWS Publishing, Boston.
Rimon, Y., and Cheng, S. I.(1969). “Numerical solution of a uniform flow over a sphere at intermediate Reynolds numbers.” Phys. Fluids, 12(5), 949–959.
Roos, F. W., and Willmarth, W. W.(1971). “Some experimental results on sphere and disk drag.” AIAA J., 9, 285–291.
Schmiedel, J.(1928). “Experimentelle untersuchungen uber die fallbewegung von kugeln und scheiben in reibenden flussigkeiten.” Phys. Z., 29, 593–610.
Shakespear, G. A.(1914). “Experiments on the resistance of the air to falling spheres.” Philos. Mag., 28, 728–734.
Stokes, G. (1880). Mathematical and physical papers, Vol. 1, Cambridge University Press, Cambridge, U.K.
Turton, R., and Clark, N.(1987). “An explicit relationship to predict spherical particle terminal velocity.” Powder Technol., 53, 127–129.
Turton, R., and Levenspiel, O.(1986). “A short note on the drag correlation for spheres.” Powder Technol., 47, 83–86.
Vlajinac, M., and Covert, E. E.(1972). “Sting-free measurements of sphere drag in laminar flow.” J. Fluid Mech., 54, 385–392.
Wieselsberger, C.(1922). “Weitere feststellungen uber die gesetze des flussigkeits und luftwiderstandes.” Phys. Z., 23, 219–224.
Wieselsberger, C.(1923). “Der widerstand von kugeln, ellipsoiden und scheiben.” Ergebn. der Aerodynamischen Versuchsanstalt, 2(28), 28–32.
Zigrang, D., and Sylvester, N.(1981). “An explicit equation for particle settling velocities in solid–liquid systems.” AIChE J., 27(6), 1043–1044.

Information & Authors

Information

Published In

Go to Journal of Environmental Engineering
Journal of Environmental Engineering
Volume 129Issue 3March 2003
Pages: 222 - 231

History

Received: Jul 27, 2001
Accepted: Feb 8, 2002
Published online: Feb 14, 2003
Published in print: Mar 2003

Permissions

Request permissions for this article.

Authors

Affiliations

Phillip P. Brown
Captain, United States Air Force, Doctoral Student, Dept. of Civil Engineering, ECJ 8.6, Univ. of Texas at Austin, Austin, TX 78712.
Desmond F. Lawler
W. A. Cunningham Professor of Engineering, Dept. of Civil Engineering, ECJ 8.6, Univ. of Texas at Austin, Austin, TX 78712 (corresponding author).

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited by

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share