Lecture 7

Setting and sedimentation: Part 1

A particl

dum F

dt

Where, m

velocity

accelerat

=rw2 for

force and

F

F

W

respectiv

particles

A

F

=g), (du/

method i

u

e settling in

e D bF F F

m is the m

of the par

tion force, a

settling und

d Fb is the bu

2f

D D

uF C

2

fb e

p

F m a

Where, CD i

vely. AP is p

having diam

2P

P

DA ,

4

or particles

/dt)=0. Putti

is given as:

tp D

2mgu

A C

PA

a fluid expe

mass of the

rticle in the

ae =g for Gr

der Centrifu

uoyancy forc

2

pA

is the drag

projected are

meter (DP), v

P

3Dm

6

settling with

ing the valu

P f

P f

ARTICLE S

eriences follo

particle, u

e fluid, eF

ravitational

ugal action.

ce and they a

coefficient,

ea of the pa

value of AP a

p

h terminal v

ues of differ

SETTLING

owing force

is the settli

ema is

settling and

FD is the dr

are given as:

, ρf and ρp

article and m

and m is give

velocity (ut) u

rent forces,

THEORY

balance:

ing

the

d ae

rag

:

are the de

m is the ma

en as:

under the fo

the termina

ensity of flu

ass of partic

orce of gravi

al velocity (

(3.7.1)

(3.7.2)

uid and par

cle. For sphe

(3.7.3)

itational forc

(ut) by New

(3.7.4)

rticle,

erical

ce (ae

wton’s

pP ft

f D

gDρ -ρ4u =

3 ρ C (for Spherical particle) (3.7.5)

Variation of CD (Drag-coefficient)

In laminar zone, Stoke’s law is applicable

f t PD

f

u D24C ; 0.01 Re 0 .1

Re

(3.7.6)

2p f P

tf

g( )Du

18

(3.7.7)

For transition zone, 0.1 Re 1 000

D n 0.6

a 18.5C

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]:

f t P

f

u DRe

(3.7.9)

pP ft

f D

gDρ -ρ4u =

3 ρ C (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:

D

24 3C 0.34

Re Re

Solution: 6f fKinematic viscosity μ 1.003 10

-6 3 -3fμ =1.003×10 ×10 =1.003×10 kg m s

Settling velocity using stokes law is:

2-32p f P

t -3f

9.81× 2.1-1 ×1000 × 1×10g( )Du 0.597 m/sec

18 18×1.003×10

3 3

f t P3

f

10 0.597 1 10u DRe 0.90 =536.32

1.003 10

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:

D

24 3C 0.34=0.5142

Re Re

pP ft

f D

gDρ -ρ4u = =0.1763 m s

3 ρ C

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.