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

di-1 - di mi μ3,i Q3,i Δ di q3,i dm,i dm i i, ,⋅ μ3

100μ3

100,

,

i

m id ⋅

11003

3

dm i

i

,

,⋅μ

∑= ⋅

μn

1i3

i,m

i,3

100d

Q0(d) q0(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

1. Seminar: Particle size distribution 2

© Dr. Werner Hintz

1) Calculation of the cumulative particle size distribution Q3(d)

( ) ( ) ( )∫=o

u

d

d33 dddqdQ

to sum up numerically in discrete intervals

( )

( ){ ( )

{Q dd

di

i

q d

i

d di

n

33

1

3

= ⋅=∑

μ ,

ΔΔ

if μ3,ii

ges

mm= - mass fraction,

Δd d di 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 d1 ≤ d2 is Q(d1) ≤ Q(d2),

• steady,

• scaling :

for d ≤ du : Q3(d) = 0 lower particle size limit

for d ≥ do : Q3(d) = 1 upper particle size limit

Calculation of the particle size frequency distribution q3(d)

( )( )

( )q ddQ dd d3

3=

numerically in discrete interval

( ) ( ) ( )q d d

Q d Q dd d di ii i

i i

i

i3 1

3 3 1

1

3−

−

−

=−−

=... ,μΔ

2) normal and log – normal diagram of Q3(d) and q3(d) : see pictures

1. Seminar: Particle size distribution 3

© Dr. Werner Hintz

3) Calculation of the median particle size d50

read from the graphical diagram of Q3(d) : d50 = 0,296 mm

Calculation of the modal particle size dh

read from the graphical diagram of q3(d) : dh = 0,175 mm

4) Calculation of the mean particle size dm,3

( ) ( ) ( )d M dq d d dm r r rd

d

u

o

, = = ∫1

for a distribution related to the quantity mass r = 3

( ) ( ) ( )d M dq d d dmd

d

u

o

,3 31

3= = ∫

in numerically form

d dm m i ii

N

, , ,3 31

= ⋅=∑ μ if the mean interval diameter is d

d dm i

i i, =

+−1

2

see exercise sheet : dm,3 = 0,463 mm

5) from the graphics of Q3(d) in a logarithmical probability diagram

μln, ,ln ln , ,3 50 3 0 296 1 217= = = −d

( )

( )σ ln, ln ln

,,

,384 3

16 3

12

12

0 6290 128

0 796= = =dd

graphical picture of Q3(d) in a RRSB – diagram

- linear correlation, curve is snapping off in the upper particle size range, not considered

′ = =d d63 0,387 mm

n = 184, by parallel displacement

n = = °=tan tan , ,α 53 5 1 35 determination of the slope angle --- deviation ???

1. Seminar: Particle size distribution 4

© Dr. Werner Hintz

using scale A dS V K, , ⋅ ′⎛

⎝⎜

⎞⎠⎟

1000for calculating surface area

i.e. specific surface area related to volume

( )

AA d

d mm mS V KS V K

, ,, , ,

,,

,=

⋅ ′ ⋅

′=

⋅=

⋅⋅ −

1000 1000 0 0107 100 387

0 0107 100 387 10

3 3

3

Amm

cmcmS V K, , ,= =27649 276 49

2

3

2

3

for Quarzit is ρs

kgm

= 2650 3

AA m

kgS m KS V K

s, ,

, , ,= =ρ

10 42

Calculation of the Sauter - diameter dST and the specific surface area related the mass AS,m

volume equivalent spheres

dV

ASTS K

=⋅6

, ⇒

→ monodisperse particle collective ⇓

with equal specific surface area

like real polydisperse particles

d

dmm

mmSTi

m ii

N= = =

=

−

∑1 1

4 940 202

3

1

1μ ,

,

,,

→ see exercise sheet

characteristic particle sizes :

∗ median particle size d50,3 = 0,296 mm

∗ modal particle size dh,3 = 0,175 mm

∗ mean particle size dm,3 = 0,464 mm

∗ Sauter - diameter dST = 0,202 mm

⇒ values of particle sizes are different !

dST

1. Seminar: Particle size distribution 5

© Dr. Werner Hintz

specific surface area

A fd dS V

ST A ST, = ⋅ =

⋅1 6

ψ with ψ A ≈ 1 for spheres

Ad

mmmmS V

i

m ii

N

,,

,,= ⋅ = ⋅ =

=

−∑6 6 4 94 296403

1

12

3

μ

respectively:

AA m m

m kgmkgS m

S V

s,

, ,= =⋅

=ρ29640

265011 2

2 3

3

2

in a good accordance with AmkgS m, ,= 10 9

2

, see RRSB - diagram

7) Calculation of Q0(d) and q0(d)

( )( ) ( )

( ) ( ) ∑∑

∫∫

=−

=−

−

−

⋅

⋅== N

i i,i,m

n

i i,i,md

d

d

d

d

d

dddqd

dddqddQ

o

u

u

1 33

1 33

33

33

0μ

μ

i = 1...n...N n – running number of intervals

N – overall number of intervals

( )( )

( )( ) ( )

q ddQ dd d

Q d Q dd

i i

i0

0 0 0 1= =− −

Δ

see working sheet

logarithmical probability diagram

μln, ,ln ln , ,0 50 0 0 022 3 82= = = −d mm

σ ln,,

,ln ln

,,0

84 0

16 0

12

12

507 6

0 942= =μμ

=dd

mm

not exact σ σln, ln, ,0 3 0 882= = caused by numerical deviations

− number distributions are shifted to the left, i.e. a lot of fine particles,