Transcript Bez tytułu slajdu - Politechnika Wrocławska

SCREENING
Screen
Feed
Screen surface
dt
(aperture)
Undersize
Oversize
Main feature of screening is size of particle
c
a
d
b
spherical particle
Irregular particle
A measure of size is usually diameter
width
length
Martin’s diameter
Feret’s diameter
equivalent perimeter
equivalent area of projection
powierzchnia rzutu
Table 4.1. Diameter of irregular particles
Name
Arithmetic diameter
Geometric diameter
Harmonic
Sieve diameter
Sieve diameter
Surface diameter
Volume diameter
Projected area
diameter
Drag diameter
Fee-falling diameter
Stokes’ diameter
Specific surface
diameter
Feret’s diameter
Martin’s diameter
Basis
Arithmetic mean of three sizes (length, width and height) of particle
((a+ b +c)/3))
Geometric mean of three sizes of particle (abc)1/3
Harmonic mean of three sizes of particle {1/3(1/a+1/b+1/c)}–1
With of a minimum square opening through which particle will pass
Arithmetic mean of two screen apertures: through which a particle
will not pass and through which particle the particle will pass
Diameter of a sphere having the same surface area as the particle
Diameter of a sphere having the same volume as the particle
Diameter of a sphere having the same projected area as the particle
when viewed in a direction perpendicular to a plane of stability
Diameter of a sphere having the same resistance to motion as the
particle in a fluid of the same viscosity and velocity when Re is small
Diameter of a sphere having the same falling speed as the particle in
the same fluid
Free-falling diameter in the laminar flow regime, Recząstki < 0,2
Diameter of a sphere having the same ratio of surface area to volume
as the particle
Mean of distance between pairs of parallel tangents to the projected
outline of the particle
Mean chord length of projected outline of particle
Shape of irregular particles ((a – length, b – width, c – height) according to Zingga
Particle shape
Spherical
Column-like
Flat-like
Flat- and column-like
Indices
b/a > 2/3; c/b > 2/3
b/a < 2/3; c/b < 2/3
b/a > 2/3; c/b < 2/3
b/a < 2/3; c/b <2/3
Table 4.3. Selected shape factors
Shape factors
Description
Formula
s =
Superficial
ratio of surface area of particle and area
calculated from nominal diameter of particle
Volumetric
ratio of volume of particle and its volume
calculated from nominal diameter of particle
s
v
Spherical

Krumbein’s
sphericity, k
Schiel’s sphericity
S
ds2/dn2
ds – diameter of sphere
having the same area as the
particle
v = (/6)dv3/(/6)dn2
dv – diameter of sphere
having the same volume as
the particle
ratio of surface area of sphere having the same
volume as the particle and the surface area of
 = (dv /ds)2
particle
ration of volume of ellipsoid (triaxial) to volume
k = {(/6)abc/(/6)a3}1/3
of sphere outlining the ellipsoid
ratio of average particle determined with circular
opening sieves (do = do 0,5) to average particle
S = (1 – log k/log 2 )
determined by squared opening sieves (dk = dk
0,5), that is (k = do /dk)
A - histogram (frequency curve)
Size distribution
size fraction content,  , %
25
20
15
10
5
0
0
200
400
600
800
1000
size fraction, di-1 di, µm
undersize
fraction content,   , %
30
20
oversize
10
0
0
200
400
600
fraction, 0,5(ci-1+ ci )
800
1000
100
size
80
60
under
cumulative fraction content
having feature value of 0 c   , %
B - distribution curve (cumulated frequency curve)
40
20
oversize
0
0
200
400
600
800
feature value, c
1000
Table 2.19. Selected functions applied for linearization of size distribution curves. c denotes value of
the feature, n is a constant, n = 1+6/s. After Kelly and Spottiswood
Name
Rosin–Rammler or Weibull
Gates–Gaudin–Schumann
Broadbent–Callcott
Gaudin–Meloy
Log-probability
 (%)/100% =
(cumulative content (%)/100% of fraction
for a given c)
1 – exp[–(c/c*)s]
[c/c*]n
1 – exp[–(c/c*)]/(1 – exp(–1)
1 – [1 – (c/c*)]n
erf [ln(c/c*)/], erf – error function, –
geometric standard deviation
Meaning of c*
c value at  = 0.632
maximum c value
maximum c value
maximum c value
median c value
Analysis
( , , , , feature, indices)
Physics
(Yn= f( X1, X2, Xn))
Screening
Mechanics
others
Screen
 F=O)
Feed
Screen surface
dt
(aperture)
Undersize
Thermodynamics
 E=O)
Oversize
Probability
(P=P 1 P2...Pn)
Mechanics of screening
Py
P
T
 + 
= 
Tx
P
Px

Gx
G
Gy
P - inertia
G- gravity
T - friction
G
Physics of screening
sliding
Relationships between forces and screening parameters:
Px = P cos ( +  ) = ma cos ( +  )
Gx = G sin  = mg sin 
Tx = o (Gy – Py) = o [mg cos  – ma sin ( +  )]
Their combination (for Px + Gx > Tx) gives:
particle sliding index us
uo – screen dynamic index
us 
u o cos (   )  u o sin (   ) 
1
u o cos   sin 
a  A
uo 
g
2
A–harmonic vibration amplitude, –angular velocity, m–particle mass, g–
accelaration due to gravity, a–accelaration of sliding, –screen slope, – angle of
inertia force
Physics of screening
casting
Casting: Py > Gy.
Py = P sin ( + )
Gy = G cos 
(G = mg, P = ma, a = A2)
casting index up
A 2 sin(   )
A 2 sin(Σ )
up 

1
g cos 
g cos 
 - ejection angle ( + ). Maximum up is 3,3 It depends on the type of screen.
Movevent of particle with casting (ejection)
a
P

difficult grain


Gy G
screen aperture dt
b
d


u
ad
screen
surfac
e
Probability of screening
Pscrening = Pz Ps Pr
Pr
way of
screening
H - thickness of particles
on screen
H
dt /H
dr - screen aperture
(1d
s=
f (l,ł,al,ał,dt)
/d t 2
)
Pscreening
screen
particle
Ps

ad A  b)
Pprzesiania  Pz Ps Pr  1 

C AN d t

d
Pz
2

rd t2
dt

 (rd t  al )( d t  ał ) H

Kinetics of screening
Generally:
vp i = –di /dt = ki i,
 - velocity of screening
i,t = i exp (–ki t)
 - content of a fraction in
the product on the screen
k - rate constant, t - time
For instance Malewski, 1990:
  di
k si  k0,5 21 
  dt
k0.5 -rate constant for di = dt




dt -screen aperture
 - constant
di - particle size
k0,5 = 3600 VBWsCds/Qo,
(for meaning of symbols see „Podstawy mineralugii, 2001”)
Table 4.4. Selected approximate formulas for output of a continuous screening
Author
Nawrocki
Kluge
Olewski
Formula
Source
Q = 900Fn0,5sdtuvm C/Sb (Mg/h)
Banaszewski, 1990
Q = FQjWgWd SHM (Mg/h)
Banaszewski, 1990
Q = 2,23·10–4 (100 – ) dt pc u (Mg/h)
Sztaba, 1993
For Olewski formula: pc - working surface area of screen,  - recovery of undersize, dt - screen
aperture, u - density of the feed
Particle size analysis
Table 2.22. Results of particle size analysis of a sample
Size fraction
m
Average particle size
m
Size fraction content
in the sample
ki = di–1di
0–125
125–160
160–200
200–250
250–320
320–400
400–500
500–630
630–800
800–1000
d = 0,5(di–1 + di)
62,5
142
180
225
285
360
450
565
715
900
i, %
0,01
2,45
12,32
24,05
21,16
13,12
11,45
7,78
3,86
3,80
cumulative content of
size fractions in the
sample
i, %
0,01
2,46
14,78
38,83
59,99
73,11
84,56
92,34
96,20
100,0
Table 2.21. Aperture of analytical sieves according to Polish Standards (PN-86/M-94001) for particle size analysis
APERTURE
mm
APERTURE
mm
Modulus ~1,12
Modulus ~1,12
Modulus ~1,26
Modulus ~1,26
APERTURE
mm
0,028
0,040
0,36
0,045
3,6
4,5
5,0
5,6
(6,0),6,3
0,71
0,80
7,1
9,0
10,0
11,2
12,0, 13,5
1,4
14,0
1,8
140
160
18,0
20,0
2,2
112
(120),125
16,0
2,0
90
100
1,1, 1,2
1,6
71
80
0,90
(1,25)
56
(60),63
8,0
1,0
45
50
0,56
0,63
36
40
0,45
0,056
28
(30), 32
4,0
0,50
0,063
0,071, (0,075)
0,080
0,090
0,10
0,11
0,12
0,14 (0,15)
0,16
0,18
0,20
0,22
2,8
(3,0), 3,2
0,40
0,050
25
0,28
0,036
cont.
2,5
0,32
Modulus ~1,26
Modulus ~1,26
cont.
0,25
0,032
Modulus ~1,26
Modulus ~1,26
cont.
0,025
Modulus ~1,12
Modulus ~1,12
Modulus ~1,26
Modulus ~1,26
APERTURE
mm
180
200
22,0,(22,4)
Basic screen types and their classsification
(Kelly and Spootiswood)