Page 387 - Water and wastewater engineering

10-4 WATER AND WASTEWATER ENGINEERING

The drag coefficient takes on different values depending on the flow regime surrounding the
particle. The flow regime may be characterized qualitatively as laminar, turbulent, or transi-
tional. In laminar flow, the fluid moves in layers, or laminas, with one layer gliding smoothly
over adjacent layers with only molecular interchange of momentum. In turbulent flow, the
fluid motion is very erratic with a violent transverse interchange of momentum. Osborne
Reynolds (1883) developed a quantitative means of describing the different flow regimes using
a dimensionless ratio that is called the Reynolds number. For spheres moving through a liquid
this number is defined as

R ()dv s (10-9)

where R Reynolds number
d diameter of sphere, m
v s velocity of sphere, m/s
2
v kinematic viscosity, m /s /
3
density of fluid, kg/m

dynamic viscosity, Pa · s
Thomas Camp (1946) developed empirical data relating the drag coefficient to Reyn-
olds number ( Figure 10-2 ). For eddying resistance for spheres at high Reynolds numbers
4
( R > 10 ), C D has a value of about 0.4. For viscous resistance at low Reynolds numbers ( R < 0.5)
for spheres:
24
C D (10-10)
R
4
For the transition region of R between 0.5 and 10 , the drag coefficient for spheres may be
approximated by the following:

24 3
C D 12 0 34 (10-11)
.
/
R R
Sir George Gabriel Stokes showed that, for spherical particles falling under laminar (quiescent)
conditions, Equation 10-8 reduces to the following:

g( s d ) 2
v s (10-12)
18

where dynamic viscosity, Pa · s

18 a constant
Equation 10-12 is called Stokes’ law (Stokes, 1845). Dynamic viscosity (also called absolute vis-
cosity ) is a function of the water temperature. A table of dynamic viscosities is given in Appendix
A. Stokes’ law is valid for spherical particles and laminar flow (Reynolds numbers 1).
The customary calculation procedure for Type I particles is to assume laminar conditions
and to use Stokes’ law to calculate a settling velocity. The Reynolds number is then checked
using this velocity. If the Reynolds number is 1, the calculation is complete. If the Reynolds