Upload
anonymous-uurfssl
View
232
Download
0
Embed Size (px)
Citation preview
8/17/2019 Particle Size Distribution Example
1/13
1. Seminar: Particle size distribution 1
© Dr. Werner Hintz
Exercise sheet:
particle size
fraction
mass mass fraction cumulative
fraction
interval
width
frequency
distribution
mean
interval
diameter
i d i-1 - d i mi μ3,i Q3,i Δ d i q 3,i d m,i d m i i, ,⋅ 3100
3
100
,
,
i
m id ⋅
1
10033
d m i
i
,
,
⋅
μ
∑= ⋅
μn
1i3
i,m
i,3
100d
Q0(d) q 0(d)
[mm] [g] [%] [%] [mm] [%mm-1] [mm] [mm] [mm-1] [mm-3] [mm-3] [-] [mm-1]
1 0-0,04 2,27 1,20 1,20 0,04 29,98 0,020 0,00024 0,6 1500 1500 0,797 20
2 0,04-0,063 5,31 2,80 4,00 0,023 121,91 0,0515 0,00144 0,544 205,0 1705,0 0,906 4,74
3 0,063-0,1 10,79 5,70 9,70 0,037 154,03 0,0815 0,00465 0,699 105,3 1810,3 0,962 1,51
4 0,1-0,25 64,02 33,80 43,50 0,15 225,34 0,175 0,0592 1,932 63,1 1873,4 0,996 0,22
5 0,25-0,4 40,34 21,30 64,80 0,15 141,99 0,325 0,0692 0,655 6,20 1879,7 0,999 0,02
6 0,4-0,63 36,56 19,30 84,10 0,23 83,93 0,515 0,0995 0,375 1,41 1881,0 1,000 0
7 0,63-1,0 13,44 7,10 91,20 0,37 19,18 0,815 0,0579 0,087 0,13 1881,1 1,000 0
8 1,0-2,5 15,34 8,10 99,30 1,5 5,40 1,750 0,142 0,046 0,02 1881,1 1,000 0
9 2,5-6,0 1,33 0,70 100,00 3,5 0,20 4,250 0,0298 0,002 0 1881,1 1,000 0
Σ 189,40 100,00 4,94 1881,1 1881,1
8/17/2019 Particle Size Distribution Example
2/13
1. Seminar: Particle size distribution 2
1) Calculation of the cumulative particle size distribution Q3(d)
( ) ( ) ( )∫=o
u
d
d
33 d d d qd Q
to sum up numerically in discrete intervals
( )
( ){ ( ){
Q d d
d i
i
q d
i
d d i
n
3
3
1
3
= ⋅=
∑ ,Δ Δ
if μ 3,ii
ges
mm= - mass fraction,
Δd d d i i i
= − −1 - interval width,
( )Q d ii
n
3 31
==
∑μ , summation from i=1...n...N
N – overall number of the intervals
→ results : see exercise sheet
Distribution functions are :
•
monotone not decreasing, i.e. for d 1 ≤ d 2 is Q(d 1) ≤ Q(d 2),• steady,
• scaling :
for d ≤ d u : Q3(d) = 0 lower particle size limit
for d ≥ d o : Q3(d) = 1 upper particle size limit
Calculation of the particle size frequency distribution q 3(d)
( ) ( )
q ddQ d 3=
8/17/2019 Particle Size Distribution Example
3/13
1. Seminar: Particle size distribution 3
3) Calculation of the median particle size d 50
read from the graphical diagram of Q3(d) : d 50 = 0,296 mm
Calculation of the modal particle size d h
read from the graphical diagram of q 3(d) : d h = 0,175 mm
4) Calculation of the mean particle size d m,3
( ) ( ) ( )d M dq d d d m r r r d
d
u
o
, = = ∫1
for a distribution related to the quantity mass r = 3
( ) ( ) ( )d M dq d d d md
d
u
o
,3 3
1
3= = ∫
in numerically form
d d m m i ii
N
, , ,3 31= ⋅=∑
μ if the mean interval diameter is d
d d
m i
i i
, =
+−12
see exercise sheet : d m,3 = 0,463 mm
5) from the graphics of Q3(d) in a logarithmical probability diagram
ln, ,ln ln , ,3 50 3 0 296 1 217= = = −d
( )σ ln ln ln
,,3
84 31 1 0 6290 796= = =
d
d
8/17/2019 Particle Size Distribution Example
4/13
1. Seminar: Particle size distribution 4
using scale A d S V K , , ⋅ ′⎛ ⎝ ⎜ ⎞
⎠⎟
1000for calculating surface area
i.e. specific surface area related to volume
( ) A
A d
d mm mS V K
S V K
, ,
, , ,
,
,
,=
⋅ ′ ⋅
′ =
⋅=
⋅⋅ −
1000 1000 0 0107 10
0 387
0 0107 10
0 387 10
3 3
3
Am
m
cm
cmS V K , , ,= =27649 276 49
2
3
2
3
for Quarzit is ρ skg
m= 2650 3
A A m
kgS m K
S V K
s
, ,
, ,,= =
ρ
10 42
Calculation of the Sauter - diameter d ST and the specific surface area related the mass AS,m
volume equivalent spheres
d V
AST S K =
⋅6
,
⇒
→ monodisperse particle collective ⇓
with equal specific surface area
like real polydisperse particles
d
d
mm mmST i
m ii
N = = ==
−
∑
1 1
4 94 0 2023
1
1μ ,
,
, ,
→ see exercise sheet
d ST
8/17/2019 Particle Size Distribution Example
5/13
1. Seminar: Particle size distribution 5
specific surface area
A f d d
S V
ST A ST
, = ⋅ = ⋅1 6
ψ with
A ≈ 1 for spheres
Ad
mmm
mS V
i
m ii
N
,
,
,
,= ⋅ = ⋅ ==
−∑6 6 4 94 2964031
1
2
3
μ
respectively:
A A m m
m kg
m
kgS mS V
s,
, ,= =⋅
= ρ
29640
265011 2
2 3
3
2
in a good accordance with Am
kgS m,,= 10 9
2
, see RRSB - diagram
7) Calculation of Q0(d) and q 0(d)
( )( ) ( )
( ) ( ) ∑∑
∫
∫=
−
=
−
−
−
⋅
⋅==
N
i i ,i ,m
n
i i ,i ,m
d
d
d
d
d
d
d d d qd
d d d qd d Q
o
u
u
1 3
3
1 3
3
3
3
3
3
0
μ
μ
i = 1...n...N n – running number of intervals
N – overall number of intervals
( ) ( )
( )
( ) ( )q d
dQ d
d d
Q d Q d
d
i i
i
0
0 0 0 1= = − −
Δ
see working sheet
logarithmical probability diagram
ln, ,ln ln , ,0 50 0 0 022 3 82= = = −d mm
d
8/17/2019 Particle Size Distribution Example
6/13
8/17/2019 Particle Size Distribution Example
7/13
8/17/2019 Particle Size Distribution Example
8/13
8/17/2019 Particle Size Distribution Example
9/13
8/17/2019 Particle Size Distribution Example
10/13
8/17/2019 Particle Size Distribution Example
11/13
8/17/2019 Particle Size Distribution Example
12/13
8/17/2019 Particle Size Distribution Example
13/13