Science of Water : Concepts and Applications

Water Treatment Calculations 361
The values of drag coeffi cient depend on the density of water ( p), relative velocity (u), particle
diameter (d), and viscosity of water ( μ), which gives the Reynolds number R as:
R pud
(10.60)
As the Reynolds number increases, the value of C D increases. For R < 2, C D is related to R by
the linear expression as follows:
CD
24
R
(10.61)
At low levels of R, the Stokes’ equation, for laminar fl ow conditions, is used (Equations 10.60
and 10.61 substituted into 10.59).
g( p) d
u
18
2
(10.62)
In the region of higher Reynolds numbers (2 < R < 500–1000), C D becomes (Fair et al., 1968):
CD
24 3
034
. (10.63)
R R
√ Key Point: In the region of turbulent fl ow (500–1000 < R < 200,000), the C D remains approximately
constant at 0.44.
The velocity of settling particles results in Newton’s equation (AWWA & ASCE, 1990):
⎡(
ppp) gd⎤
u 1.74
⎢
⎣ p
⎥
⎦
05 .
(10.64)
√ Key Point: When the Reynolds number >200,000, the drag force decreases substantially and
C D becomes 0.10. There is no settling at this condition.
Example 10.65
Problem:
Estimate the terminal settling velocity in water at a temperature of 21°C of spherical particles with
specifi c gravity 2.40 and average diameter of (a) 0.006 mm and (b) 1.0 mm.
Solution:
Step 1. Use Equation 10.62 for (a).
Given: At temperature (T) = 21°C
ρ = 998 kg/m3 μ = 0.00098 N s/m2 d = 0.06 mm = 6 × 10 –5 m
g = 9.81 m/s2