particle size distribution example

Upload: anonymous-uurfssl

Post on 06-Jul-2018

232 views

Category:

Documents


0 download

TRANSCRIPT

  • 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

    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 

    u

    o

    ,   = = ∫1  

    for a distribution related to the quantity mass r = 3

    ( ) ( ) ( )d M dq d d d  md 

    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= = =

  • 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

    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 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  

    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