# Sedimentation of Particles in Water or Air

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22 Φεβ 2014 (πριν από 4 χρόνια και 4 μήνες)

121 εμφανίσεις

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1

P.
Hoffman

Datum tisku:
22.02.

Sedimentation of Particles in Water or Air

Example of WWT exercises

Uniform straight
-
line motion

The case is for example a particle sedimentation in the gravitational field
when the particle has a constant falling speed.

Balance of forces act to a pa
rticle

-

Steady speed motion is important for our

solution (particles sedimentation).

-

A particle moves with a constant falling

speed u
SC
. Forces acting to the particle

have to be in balance (there is not

acce
l
eration or deceleration).

-

The steady speed is reached for time

†
i渠n牡xis⁦湡l⁳e敤†甠㴠〬u9*u
SC

is used.

We suppose a spherical particle with following acting forces:

Gravitational force

g
d
G
P
*
*
6
*
3

Lifting force

g
d
F
LF
*
*
6
*
3

D
rag force

*
2
*
4
*
*
2
2
S
D
D
u
d
C
F

Inertial force

F
IN

= 0

d

P

,

G

F
LF

F
D

u
SC

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2

P.
Hoffman

Datum tisku:
22.02.

The forces have to be in balance

G = F
LF

+ F
D

g
d
P
*
*
6
*
3

=
g
d
*
*
6
*
3

+

*
2
*
4
*
*
2
2
S
D
u
d
C

*
*
)
(
*
*
3
4
2
D
P
S
C
g
d
u

Value of the d
rag coefficient C
D

depends on Re number and thereby on u
n-
known sedimentation speed u
S

too
.
Therefore it is impossible to set the coeff
i-
cient. But we can use relations valid for laminar and turbulent regions of part
i-
cles sediment
a
tion.

-

For laminar (Stok
es) region it is valid:

S
D
C
Re
24

where

3
,
0
2
,
0
*
*
Re

d
u
S
S

..... for accuracy

-

0ⰵ‥

†††††

2‮⸮⸮⸮⸮⸮⸮⸮⸮.⁦潲⁡捣畲ac†

-

5‥

-

F爠瑵牢rl敮琠⡎ewt温⁲ngi渠n琠ts⁶ali携

C
D

= 0,44

3*10
5

S

400

500

-

For transient region it is valid:

C
D

= 18,5 * Re
S
-
0,6

or

6
Re
1
*
Re
24
3
/
2
S
S
D
C

After a substitution we can set the
sedimentation speed of the particle in the
la
m
inar region:

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3

P.
Hoffman

Datum tisku:
22.02.

*
*
24
*
*
*
*
)
(
*
*
3
4
2
d
u
g
d
u
S
P
S

and after modific
a
tion
it is

*
18
*
)
(
*
2
g
d
u
P
S

Analogously after substitution we can set the
sedimentation speed of the particle
in the turbulent region:

*
44
,
0
*
)
(
*
*
3
4
2
g
d
u
P
S

and after mod
i
fication it is

g
d
u
P
S
*
)
(
*
*
74
,
1

-

Setting of a sedi
mentation region

A dimensionless term is used for this setting. The term contains only known va
l-
ues, it is parameters of the particle and its ambient (water, air etc.).

2
3
2
*
*
)
(
*
*
3
4
Re
*

g
d
C
P
S
D

For L region it is valid

C
D
*Re
S
2

12

48 (accuracy +/
-

0,5

5 %)

For T region it is valid

1,1*10
5

C
D
*Re
S
2

4*10
10

Note:

Sludge and fine sand particles usually settle in the laminar region

b礠e慳潮⁯

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4

P.
Hoffman

Datum tisku:
22.02.

Ex. 1: Sedimentation of sludge particles in water.

Given: Sludge d = 0,2 mm;

p

= 1020 kg/m
3
;

㴠=98⁫g/m
3
;

㴠=*10
-
3

Pa*s

Calculate a sedimentation speed u
S

= ?, time

㴠=w桥渠h桥⁰慲hicl攠

††

㴠=
ⰹ9⁵
S

and a distance h = ? that is covered during

the time

.

Setting of a sedimentation region

48
12
30
,
2
)
10
*
1
(
81
,
9
*
998
*
)
998
1020
(
*
)
10
*
2
,
0
(
*
3
4
Re
*
2
3
3
3
2

S
D
C

䰠牥杩潮

Setting of the sedimentation speed of the particle

h
m
s
m
u
S
/
73
,
1
/
10
*
80
,
4
10
*
1
*
18
81
,
9
*
)
998
1020
(
*
)
10
*
2
,
0
(
4
3
2
3

Setting of the time when the part
icle reaches the sedimentation speed u

0ⰹ9*u
S

)
1
ln(
*
*
18
*
2
S
P
u
u
d

s
2
3
2
3
10
*
04
,
1
)
99
,
0
1
ln(
*
10
*
1
*
18
1020
*
)
10
*
2
,
0
(

Setting of the distance that the particle covers during the time

)
1
ln(
*
18
Re
*
*
S
S
S
č
u
u
u
u
d
h

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5

P.
Hoffman

Datum tisku:
22.02.

mm
m
h
3
6
3
10
*
93
,
3
10
*
93
,
3
)
99
,
0
1
ln(
99
,
0
*
18
0958
,
0
*
10
*
2
,
0
*
998
1020

where

2
0958
,
0
10
*
1
998
*
10
*
2
,
0
*
10
*
80
,
4
*
*
Re
3
3
4

d
u
S
S

Simi
lar calculations are for sedimentation of gypsum or sand.

Ex. 2: Sedimentation of gypsum particles in water.

G:

d = 0,1 mm;

P

= 1800 kg/m
3
;

㴠=98⁫g/m
3
;

㴠=*10
-
3

Pa*s

T:

Calculate a sedimentation speed u
S

= ?, time

㴠=w桥渠瑨攠pa牴i捬攠

㴠=ⰹ9⁵
S

and a distance h = ? that is covered during

the time

.

†
48
12
5
,
10
)
10
*
1
(
81
,
9
*
998
*
)
998
1800
(
*
)
10
*
1
,
0
(
*
3
4
Re
*
2
3
3
3
2

S
D
C
†

䰠牥gin

h
m
s
m
u
S
/
74
,
15
/
10
*
37
,
4
10
*
1
*
18
81
,
9
*
)
998
1800
(
*
)
10
*
1
,
0
(
3
3
2
3

u

㴠=ⰹ9*u
S

s
3
3
2
3
10
*
6
,
4
)
99
,
0
1
ln(
*
10
*
1
*
18
1800
*
)
10
*
1
,
0
(

Setting of the distance that the particle covers during the time

mm
m
h
0158
,
0
10
*
58
,
1
)
99
,
0
1
ln(
99
,
0
*
18
436
,
0
*
10
*
1
,
0
*
998
1800
5
3

where

2
436
,
0
10
*
1
998
*
10
*
1
,
0
*
10
*
37
,
4
Re
3
3
3

S

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6

P.
Hoffman

Datum tisku:
22.02.

Ex. 3: Sedimentation of sand p
articles in water.

G:

d = 0,1 mm;

P

= 2300 kg/m
3
;

㴠=98⁫g/m
3
;

㴠=*10
-
3

Pa*s

T:

u
S

= ?,

㴠=⁦潲⁵

㴠=ⰹ9⁵
S

and h = ? for time

.

48
12
17
)
10
*
1
(
81
,
9
*
998
*
)
998
2300
(
*
)
10
*
1
,
0
(
*
3
4
Re
*
2
3
3
3
2

S
D
C
†

䰠牥杩潮

h
m
s
m
u
S
/
5
,
25
/
10
*
10
,
7
10
*
1
*
18
81
,
9
*
)
998
2300
(
*
)
10
*
1
,
0
(
3
3
2
3

u

㴠=ⰹ9*u
S

s
3
3
2
3
10
*
9
,
5
)
99
,
0
1
ln(
*
10
*
1
*
18
2300
*
)
10
*
1
,
0
(

Setting of the distance that the particle covers during the time

mm
m
h
0328
,
0
10
*
28
,
3
)
99
,
0
1
ln(
99
,
0
*
18
709
,
0
*
10
*
1
,
0
*
998
2300
5
3

w桥牥h

2
709
,
0
10
*
1
998
*
10
*
1
,
0
*
10
*
10
,
7
Re
3
3
3

S

䕸⸠4㨠

.

P

= 2300 kg/m
3
;

㴠=98kg/m
3
;

㴠=*10
-
3

Pa*s

T:

u
S

= ?,

㴠=⁦潲⁵

㴠=ⰹ9⁵
S

and h = ? for time

.

†
48
6
,
46
)
10
*
1
(
81
,
9
*
998
*
)
998
2300
(
*
)
10
*
14
,
0
(
*
3
4
Re
*
2
3
3
3
2

S
D
C

䰠牥gin

⡢(畮uar⁢整w敥渠䰠慮搠瑲a湳i敮琠牥ti湳)

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P.
Hoffman

Datum tisku:
22.02.

Note:

For a result accuracy c. +/
-

5 % ..... limit of application of relations for the laminar region.

Note:

For the same conditions but sand particle diameter d = 1,9 mm we reach a begin
ning of the turbulent

region (C
D
*Re
P
2

= 1,17*10
5
).

Setting of the sedimentation speed of the particle

h
m
s
m
u
S
/
1
,
50
/
10
*
9
,
13
10
*
1
*
18
81
,
9
*
)
998
2300
(
*
)
10
*
14
,
0
(
3
3
2
3

Setting of the time when the particle reaches the sedimentation speed

u

㴠=ⰹ9*u
S

s
3
3
2
3
10
*
5
,
11
)
99
,
0
1
ln(
*
10
*
1
*
18
2300
*
)
10
*
14
,
0
(

Setting of the

distance that the particle covers during the time

mm
m
h
13
,
0
10
*
26
,
1
)
99
,
0
1
ln(
99
,
0
*
18
94
,
1
*
10
*
14
,
0
*
998
2300
4
3

w桥牥h

2
94
,
1
10
*
1
998
*
10
*
14
,
0
*
10
*
9
,
13
Re
3
3
3

S

䕸⸠5

.

P

= 2300 kg/m
3
;

㴠=ⰱ,0⁫g/m
3
;

㴠=ⰸ㈪10
-
5

Pa*s

T:

u
S

= ?,

㴠=

㴠=ⰹ9⁵
S

and h = ? for time

.

48
6
,
45
)
10
*
82
,
1
(
81
,
9
*
19
,
1
*
)
19
,
1
2300
(
*
)
10
*
075
,
0
(
*
3
4
Re
*
2
5
3
3
2

S
D
C
†

䰠牥杩潮

s
m
u
S
/
387
,
0
10
*
82
,
1
*
18
81
,
9
*
)
19
,
1
2300
(
*
)
10
*
075
,
0
(
5
2
3

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P.
Hoffman

Datum tisku:
22.02.

Setting of the time when the particle reaches the sedimenta
tion speed

u

㴠=ⰹ9*u
S

s
182
,
0
)
99
,
0
1
ln(
*
10
*
82
,
1
*
18
2300
*
)
10
*
075
,
0
(
5
2
3

Setting of the distance that the particle covers during the time

mm
m
h
3
,
55
0553
,
0
)
99
,
0
1
ln(
99
,
0
*
18
90
,
1
*
10
*
075
,
0
*
19
,
1
2300
3

I琠ts⁵獵all⁩灯ssi扬攠e⁮敧l散琠瑨攠摩s瑡湣攮

w桥牥h

2
9
,
1
10
*
82
,
1
19
,
1
*
10
*
075
,
0
*
387
,
0
Re
5
3

S

䕸⸠6㨠:us琠灡牴rcl攠s敤ime

P

= 2300 kg/m
3
;

㴠=ⰱ,0⁫g/m
3
;

㴠=ⰸ㈪10
-
5

Pa*s

T:

u
S

= ?,

㴠=⁦潲⁵

㴠=ⰹ9⁵
S

and h = ? for time

.

†
48
12
84
,
0
)
10
*
82
,
1
(
81
,
9
*
19
,
1
*
)
19
,
1
2300
(
*
)
10
*
20
(
*
3
4
Re
*
2
5
3
6
2

S
D
C

䰠牥gin

s
m
u
S
/
0275
,
0
10
*
82
,
1
*
18
81
,
9
*
)
19
,
1
2300
(
*
)
10
*
20
(
5
2
6

u

㴠=ⰹ9*u
S

s
0129
,
0
)
99
,
0
1
ln(
*
10
*
82
,
1
*
18
2300
*
)
10
*
20
(
5
2
6

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P.
Hoffman

Datum tisku:
22.02.

Setting of the distance that the particle covers during the time

mm
m
h
28
,
0
000277
,
0
)
99
,
0
1
ln(
99
,
0
*
18
0359
,
0
*
10
*
20
*
19
,
1
2300
6

w桥

2
0359
,
0
10
*
82
,
1
19
,
1
*
10
*
20
*
0275
,
0
Re
5
6

S

Note:

For the same conditions (sand particles in air) but diameter d = 1,0 mm it is C
C
*Re
P
2

= 1,08*10
5

it is that we are at the beginning of the turbulent region.

Ex.7: Free fall of a hailstone from a storm cloud

G:

d
= 20 mm;

P

= 918 kg/m
3
;

㴠=ⰱ,0⁫g/m
3
;

㴠=ⰸ2*10
-
5

Pa*s

T:

Set a falling (sedimentation) speed u
S

= ?

Setting of a sedimentation region

8
2
5
3
3
2
10
*
45
,
3
)
10
*
82
,
1
(
81
,
9
*
19
,
1
*
)
19
,
1
918
(
*
)
10
*
20
(
*
3
4
Re
*

S
D
C

I琠ts⁶ali搠爠瑨攠r敧i渠nC
D

= 0,44

Setting of the Reynolds number

4
8
2
10
*
77
,
2
44
,
0
10
*
45
,
3
Re
*
Re

D
S
D
S
C
C

Setting of the constant sedimentation speed of the particle

s
m
g
d
u
P
S
/
4
,
21
19
,
1
81
,
9
*
)
19
,
1
918
(
*
10
*
20
*
74
,
1
*
)
(
*
*
74
,
1
3

h
km
s
m
d
u
S
/
76
/
2
,
21
19
,
1
*
10
*
20
10
*
82
,
1
*
10
*
77
,
2
*
Re*
3
5
4

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10

P.
Hoffman

Datum tisku:
22.02.

Note 1:

The biggest in the Czech Republic observed hailstone had the diameter c. 120
mm. The corresponding falling spee
d is c. 52,4 m/s = 188 km/h.

Note 2:

As the falling time is relatively short the majority of the hail mass has to be
formed in clouds in rising air flows. These rising flows have to have approx
i-
mately the same speed as the falling speed is.