Lecture 7
Setting and sedimentation: Part 1
THEORY
PA
ARTICLE SETTLING
S
A particle settling in a fluid expeeriences follo
owing force balance:
m
du
 Fe  FD  Fb
dt
Where, m is the mass
m
of the particle, u is the settliing
velocity of the parrticle in thee fluid, Fe   mae  is the
accelerattion force, ae =g for Grravitational settling andd ae
=rw2 for settling und
der Centrifu
ugal action. FD is the drrag
d Fb is the bu
uoyancy forcce and they are
a given as::
force and
f u2
FD  CD
Ap
2
(3.7.1)
f
ae
p
(3.7.2)
Fb  m
Where,
W
CD is
i the drag coefficient,, ρf and ρp are the deensity of fluuid and parrticle,
respectiv
vely. AP is projected
p
areea of the paarticle and m is the maass of particcle. For spheerical
particles having diam
meter (DP), value
v
of AP and
a m is giveen as:
D P
 D 2P
p
AP 
,m 
6
4
3
(3.7.3)
For particles settling with
h terminal velocity
v
(ut) uunder the foorce of graviitational forcce (ae
=g), (du//dt)=0. Puttiing the valu
ues of differrent forces, the terminaal velocity ((ut) by New
wton’s
method is
i given as:
ut 
2 m g  P   f
A pC D  Pf

(3.7.4)
ut =
4  ρ P -ρ f  gD p
(for Spherical particle)
3 ρf
CD
(3.7.5)
Variation of CD (Drag-coefficient)
In laminar zone, Stoke’s law is applicable
CD 
ut 
24
;
Re
  u D 
0.01  Re   f t P   0 .1
f 

(3.7.6)
g( p   f )D 2P
(3.7.7)
18 f
For transition zone,
CD 
0.1 Re 1000
a
18.5
 0.6
n
Re
Re
(3.7.8)
For turbulent zone, CD is independent of Re and CD=0.4
For non-spherical particles, formula for Reynold number and settling velocity calculation are
modified using the shape factor (  ) [1]:
Re  
ut =
f ut DP
f
(3.7.9)
4  ρ P -ρ f  gD p
3 ρf
C D
(3.7.10)
Problem 3.7.1: A sand particle has an average diameter of 1 mm and a shape factor of 0.90 and a
specific gravity of 2.1, determine the terminal velocity of the particle settling in water at 20 oC
(kinematic viscosity of water=1.003×10-6 m2/s and specific gravity=1). Drag coefficient can be
computed using the following equation:
CD 
24 3

 0.34
Re Re
Solution: Kinematic viscosity    μf f 1.00310
6
μf =1.003×10-6×103 =1.003×10-3 kg m s
Settling velocity using stokes law is:
ut 
g(p  f )D 2P
18 f

9.81×   2.1-1 ×1000  × 1×10-3 
18×1.003×10-3
2
 0.597 m/sec
10 3  0.597  1  10  3 
f u t D P
Re  
 0.90
=536.32
f
1.003  10  3
Since Re>1, therefore, Newton’s law should be used for finding terminal velocity in
transition zone. For initial assumption of settling velocity, stoke’s law is used. This initially
assumed velocity is used to determine the Reynold number which is further used to find settling
velocity. This iterative procedure is repeated till initial assumed velocity is approximately equal
to settling velocity calculated from Newton’s equation.
Initial drag coefficient is calculated as:
CD 
24
3

 0.34=0.5142
Re Re
ut =
4  ρ P -ρ f  gD p
=0.1763 m s
3 ρf
C D
Now, iterative procedure is continued:
ut (previous calculated)
Re
CD
ut
Difference
0.5977
536.3272 0.5143
0.1763
0.4214
0.1763
158.2037 0.7302
0.1480
0.0283
0.1480
132.7684 0.7811
0.1431
0.0049
0.1431
128.3690 0.7917
0.1421
0.0010
0.1421
127.5052 0.7939
0.1419
0.0002
0.1419
127.3315 0.7943
0.1419
0.0000
Final settling velocity=0.1419 m/s.
REFERENCES
Metcalf & Eddy, Tchobanoglous, G., Burton, F. L., Stensel, H. D. “Wastewater engineering:
treatment and reuse/Metcalf & Eddy, Inc.”, Tata McGraw-Hill, 2003.