Granulometry - Lehrstuhl für Mechanische Verfahrenstechnik

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OTTO-VON-GUERICKE-UNIVERSITÄT MAGDEBURG
Department of Process Engineering
Chair of Mechanical Process Engineering
Practical training Granulometry I
Content:
1
Introduction
2
Problem
3
Carrying out test sieving
4
Evaluation of the experiment and discussion of the results
5
Notes for preparation to the practical training
6
Symbol index
7
References
Hintz©Praktikum Granulometry I 2007.doc;
Vorlesung "Mechanische Verfahrenstechnik"; Prof. Dr.-Ing. habil. Tomas
1
1.
Introduction
Frequently disperse systems of solids are raw material or product in the process industry
(comminution, classifying, dust collection etc.). The knowledge of the dispersed state of material is of great importance, because size and shape influence the properties of the particle
population (e.g. separation, mixing, agglomeration behaviour, flow properties).
By test sieving, sedimentation analysis and optical spectroscopy any particulate material can
be analysed. For fine particles smaller than 70 µm a sedimentation test can be applied. For an
integral characterisation of particulate material the surface area is used.
2.
Problem
By doing the experiment and evaluating the results in the practical training you should know
- the method frequently used for analysing particulate material – test sieving,
- the evaluation of test sieving and calculation of the specific surface area as well as
- the calculation of characteristic granulometrical parameter
a) A particulate material has to be split up by a sample splitter. By evaluating the results of
the done test sieving, please discuss the precision of sample splitting.
b) With the samples obtained by material splitting two independent test sieving analyses have
to be done.
By graphical tests (normal paper, logarithmic probability paper, double logarithmic diagram
etc.) please check if the experimental distributions Q3 ( d ) obey a known analytical particle
size distribution (e.g. RRSB, GGS, log normal Gauss distribution).
c) From the experimental results of the test sieving, please determine:
particle size frequency distribution q3(d)
median particle size d50
modal particle size dh
mean particle size dm
surface or SAUTER - diameter dST
volume related specific surface area A SV
d) Please convert the particle size distribution on mass basis Q 3 (d ) for one test sieving analysis to the particle size distribution on number basis Q 0 (d ) . Q 0 (d ) has to be drawn graphically (normal paper, logarithmic normal diagram, logarithmic probability paper, double
logarithmic diagram).
e) For a given material the specific surface area has to be determined by the Blaine test.
Please calculate the mean values and the standard deviation of the Blaine surfaces (2 test
series with 10 measurements).
2
3.
Carrying out test sieving
3.1 Introduction
Two identical samples of particulate material has to be sieved. Both samples are prepared by
splitting equally the starting material (about 9kg) into two fractions by giving it on a sample
splitter (see figure1). One fraction has to be split again, the other fraction is lost. The procedure has to be done until there are two fractions of about 500g. Both samples are analysed
separately. The sample mass is determined by balancing.
Figure 1: sample splitter
For sieving a sieving set is used closed with a bottom, on the sieve with the largest sieving
mesh there is a top (see figure 2). The sieving mesh is equivalent the particle size. By a vibrating sieving set generated with the help of a sieving machine the particulate material is
aerated, for particles it is possible to pass the sieve as function of the mesh size.
Figure 2: sieving set
3
Duration of sieving has to be chosen in that way that all particles which have a size smaller
than the sieving mesh are able to pass the corresponding sieve. Criterion for sieving duration
is the changing of mass fraction (oversize mass) on the sieve. Sieving can be finished when
the changing in mass fraction with time is < 0.001 g/(g overall mass . min). Abrasion resistance of the material has to be taken into account. That is why the sieving time is chosen that
abrasion and comminution, as the chase may be, by sieving of sensitive materials are disable.
3.2 Carrying out test sieving
Test sieving starts with the determination of the weight of the sample which is used. For that
the container with the sample in it is weighed. After the experiment the container with the
sample in it is weighed again to verify the sample mass used. After putting the single sieves
together to a sieving set the sample is given at the upper sieve, then sieving starts. After finishing the experiment the sieving material is filled in a given basin (test sieves has to be
cleaned with a soft brush) and is weighed (difference balancing). The oversized material
forms the particle mass mi of the class i with the grain sizes di-1 and di (di is the sieve being
above).The class of that material being on the top sieve is limited by the upper grain size d O
which has to be estimated, as the case may be.
Test sieving has to be done for two times under the same conditions. Generally there is a
small lose of mass during sieving by e.g. adhesion of finest particle. The lose should be limited to
μ
μloss
mges
m0
mi
loss
=
m −m
m
0
ges
(1)
〈0.01....0.02
0
- fraction of lost mass
- overall mass after sieving
- overall mass before sieving
- mass of the i-th class
N
m
ges
= ∑ mi
i =1
Normally the lost material is added to the smallest class.
The calculation of the granulometric parameters is shown in the following section:
With the help of the mass fraction μ 3,i of the class i
μ 3,i =
mi
m0
(2)
the cumulative particle size distribution can be determined by summing up
Q 3 (d ) =
d
n
du
i =1
∫ q 3 (d) ⋅ d (d ) = ∑ μ 3, i
(3)
4
The particle size frequency distribution is given by following equation
q 3 (d i − 1 .. d i ) =
dQ 3 (d ) μ 3, i
=
,
Δd i
d (d )
(4)
where Δd i is the class width calculated from
Δd i = d i - d i-1
q 3, i
(5)
- particle size frequency function
Δd i - class width
- mass fraction
μ 3,i
D i = Q 3, i - undersize distribution (cumulative particle size distribution)
The cumulative particle size distribution on the number basis Q 0 (d ) can be calculated using
the incomplete initial moment (particle sizes from du ....d)
M k ,r
d
d
=
du
∫ d k ⋅ q r (d ) ⋅ d (d )
(6)
du
and the complete initial moment (integration from du..do)
d0
d0
=
M k ,r
du
∫ d k ⋅ q r (d ) ⋅ d (d )
(7)
du
so that
M − 3,3
Q 0 (d ) =
d
du
M − 3,3
d
∫d
−3
du
= d
0
n μ
3, i
3
i = 1 d m, i
N μ
3, i
3
i = 1 d m, i
∑
⋅ q 3 (d ) ⋅ d (d )
=
∫ d − 3 ⋅ q 3 (d ) ⋅ d (d )
du
,
(8)
∑
The particle size frequency distribution no the number basis given by
q 0 (d i −1 ... d i ) =
Q 0 (d i −1 ) − Q 0 (d i )
.
d i −1 − d i
(9)
The equation for calculating the weighted mean diameter is
5
d0
d m,3 = M 1,3 =
∫d
1
N
⋅ q 3 (d ) ⋅ d (d ) = ∑ d m.i ⋅ μ 3,i
(10)
i =1
du
with
M1,3
r
- 1. initial moment of the quantity mass (r = 3)
⎧0 = number,1 = length,2 = surfacearea ⎫
- quantities
⎨
⎬
⎩3 = volume( mass)
⎭
where the mean class width is
(d i −1 + d i )
2
d m,i =
(11)
In the end the complete initial moment of a given quantity r can be transformed into another
one of the quantity t using:
M k ,t =
M ( k + t − r ),r
(12)
M ( t − r ),r
So, e.g. from the given mass distribution (r = 3), the k =1st complete initial moment of the
surface distribution t=2 is
M 1,2 =
M 0,3
M −1,3
.
(13)
Taking into account the normalisation condition
d0
M 0,3 = ∫ d 0 ⋅ q 3 (d )d (d ) = Q3 (d 0 ) − Q 3 (d u ) = 1 − 0 = 1
du
the SAUTER diameter as an integral parameter (equivalent sphere diameter of equal specific
surface area) is
d ST =
1
∫ d − 1 ⋅ q 3 (d ) ⋅ d (d )
=
1
N μ
3, i
(14)
∑
d
i = 1 m, i
as well as the volume related specific surface area is
AS
6
6
6 N μ 3, i
A S, V =
=
⋅ M − 1,3 =
=
⋅∑
V
ψ A ⋅ d ST ψ A i = 1 d m, i
ψA
(15)
6
ψA
- shape factor of particles (set mostly = 1)
as well as the mass related specific surface area is
A S, m =
A S A S, V
=
.
m
ρS
ρS
solid density
(16)
3. 3 Determination of the specific surface area using the Blaine test
When a fluid flows through a packing of porosity ε, there is a pressure loss Δp caused by the
packing. The mean fluid velocity in the pores is uε . For calculation the pore system is idealized as a pore system consists of n parallel cylindrical capillaries having a diameter dh and a
length 1 (equals the packing height Δhb). A measure for the pore size is the so called mean
hydraulic diameter dh (= mean capillary diameter).
dh =
4 ⋅ A durchströmt
4 ⋅ VPoren
=
=
U benetzt
A S,Poren
π 2
⋅d ⋅l
4 h
n⋅π ⋅dh ⋅l
4⋅n⋅
(17)
Considering the porosity (pore volume fraction) ε = VPoren/(VPoren + V) and the specific surface area equality of an ideal pore system and a real packing (dST-Sauter-diameter) equation
(18) results:
dh =
4 ⋅ ε ⋅ Vs
2 ⋅ ε ⋅ d ST
4⋅ε
=
=
(1- ε ) ⋅ A S
(1- ε ) A S,V
3(1- ε )
(18)
In according to physical basics for particle flow in fluids, it can be assumed that for the flow
through particle beds the pressure loss (flow resistance) consists of two effects
( Δp ∼ η ⋅ u) and
a) a viscous drag
b) a inertia drag
( Δp ∼ ρ F ⋅ u 2 )
η
fluid viscosity
fluid density
ρF
with u = ε ⋅ u ε for the local velocity of current.
Hence the pressure gradient can be introduced
grad p =
Δp
dp
=
= f(d ST , U, ε , η, ρ F )
l
dh h
(19)
as well as the Euler number
7
Eu =
Δp
(20)
ρf ⋅ u2
and the Reynolds number
Re =
U ⋅ d ST ⋅ ρ f
η
(21)
as dimensionless numbers. Equation (19) can be rearranged as follows:
Eu ⋅
d ST
= f(Re, ε )
l
(22)
For a laminar flow Re< 1 with using
kV
(1 - ε ) 2
f (Re, ε ) =
⋅
Re
ε3
kV = 180
kV = 150
(23)
monodisperse packing of spheres
grinding product of a narrow particle size distribution
2
the equation of Carman and Kozeny can also be used (Δp ∼ 1 / d ST
)
Δp
η (1 − ε ) 2
= kV ⋅ 2 ⋅
⋅u
l
d ST
ε3
(24a)
kV
Δp
(1 − ε ) 2
=
⋅ η ⋅ A S2,V ⋅
⋅u
l
36
ε3
(24b)
respectively
For doing the Blaine test the manometer liquid has to be lifted by suctioning (valve open)
from the initial rest level, then the valve has to be closed. For a manometer liquid of the denπ
sity ρl, M having the cross – sectional area d 2M and the height hM above the initial rest level,
4
the pressure affecting the packing is
Δp = 2 ⋅ ρ l,M ⋅ g ⋅ h M
If the cross sectional area of the packing is
(25)
π 2
d , then the level of the manometer liquid de4 b
creases with time
−
π 2 dh u π 2
d ⋅
= db ⋅ u
4 M dt
4
(26)
8
Hence
⎛ dM ⎞
⎟
u = -⎜
⎝ db ⎠
2
dh M
dt
(27)
Using the Carman Kozeny equation (24b) and eq. (25) one finds by separation of the variables
ρ l,M ⋅ g ⋅ d 2b
dh M
ε3
72
1
−
=
⋅
⋅
⋅
⋅ dt
hM
kV
l ⋅ η ⋅ d 2M
(1 - ε ) 2
A 2s,V
(28)
after integration from hM1 to hM2
A S2,V
ρ l,M ⋅ g ⋅ d 2b
t 2 − t1
72
ε
=
⋅
⋅
⋅
2
2
h M1
kV
l ⋅ η⋅ dM
(1 - ε )
ln
h M2
(29)
All parameters only depending on the device can be combined to an apparatus constant KG :
A S2,V
= KG ⋅
ε3
(1 − ε )
2
⋅
t 2 − t1
η
(30a)
ρ l, M ⋅ g ⋅ d 2b
72
KG =
⋅
k V l ⋅ d 2M ⋅ ln h M1 / h M 2
(30b)
where the mass related specific surface area is AS,m = AS,V/ρs .
The constant KG can be determined by comparing with a known capillary or a material of
known specific surface area. With using gases as a streaming fluid the density change can be
neglected (with Δp being only small).
If the Sauter diameter is dST < 5 µm (AS,V > 1,2 m²/cm3) the pores are so small that the average free path length λ g is not large enough for the diffusion of the gas molecules ( λ g = 0,06
µm for air at the standard pressure, θ = 20 °C, with λ g = 1/ρg). With increasing Knudsen
number (d = dh ≈ dST)
Kn =
λg
d
≈
λg
d ST
≈ A S, V ⋅ λ g / 6 > 01
,
(31)
flow resistance decreases:
9
−1
k V (1 − ε )
kV
Δp
(1 − ε ) 2
⎧
⎫
1
0
762
(32)
u
A
,
=
⋅ η ⋅ A S2, V ⋅
⋅
⋅
+
⋅
⋅
⋅
⋅
λ
⎨
S, V
g⎬
36
l
36
ε
⎩
⎭
ε3
Equation (32) characterises the fluid flow through a packing in a broad range of the Knudsen
number. Flow consists of a laminar share (continuum streaming) and a share of so called molecular streaming. That means, in that case equation (30c) is valid analogous to equation
(30a,b):
A S,V ≈ 0,762 ⋅
ρ l,M ⋅ g ⋅ d 2b
ε 2 t 2 − t1
⋅
η
l ⋅ d 2M ⋅ ln h M1 / h M 2 1 − ε
⋅
(30c)
1. Determination of the weight of the empty measuring cell with the sieving plate and 2 sheets
of filter paper (without piston)
2. Fill in the powder in portions with a funnel in the measuring cell. Compress the powder
with a special piston being longer than the original piston. The special piston has a mark to
see how deep the piston is allowed to go into the measuring cell to ensure the necessary
solid bed volume. After filling in the last portion of powder and sealing the bed surface
with the second filter paper the densification of the powder follows in the known manner
with the help of the original piston.
3. Determination of the filled measuring cell (compare2). The mass difference between this
mass and the mass at the beginning of the filling procedure (empty weight) is the mass of
the solid (m).
4. Determination of the running time of the Blaine apparatus in the known manner.
5. The viscosity of air as a function of temperature is given by the Sutherland equation:
η = 1,503 ⋅ 10 -6
T
T
in Pa ⋅ s
123,6
1+
T
temperature in K
6. Several materials show a special behaviour in that way that after compression a very strong
reverse densification occurs producing a slit between the surfaces of the piston and the
edge of the measuring cell. The slit can be determined with a gauge.
10
Figure 4:
1
2
Blaine apparatus
3
Cylinder
U-tube-manometer with bevelled
extension
Sieving plate
4
Piston
5
6
2 path-valve
3 path-valve
7
Rubber ball for producing the
necessary small vacuum
8
A
Thermometer
Suction mark of the manometer
B
C
Gauge mark of the manometer
Gauge mark of the manometer
11
the corrected value are then:
V* = V ⋅
ε
= 1-
l+s
l
Δm
ρs ⋅ V*
l
K *G = K G ⋅
l+s
7. If you use powder with known mass related surface area in the experiment you can determine the apparatus constant KG. The measured Blaine surface area for other materials
arises as values relative to the calibration material.
4.
Evaluation of the experiment and discussion of the results
4.1 Test sieving
From the data of the experiment the functions Q 3 (d ) , q 3 (d ) as well as Q 0 (d ) and
q 0 (d ) have to be calculated and represented graphically. The parameters of the distribution
given in the instruction have to be determined.
The conversion of the mass distribution in the number distribution results from the equations
(8) and (9).
For graphical analysis :
- Representation of the functions Q 3 (d ) , Q 0 (d ) , q 3 (d ) and q 0 (d ) in normal diagram
- Representation of the functions Q 3 (d ) and Q 0 (d ) as logarithmic normal distribution
(log.
Gauss function) and determination of the standard deviation σln as well as the
determination of μ ln,0 and μ ln,3
Q 3 (d ) =
with
u=
u
⎛ − t2 ⎞
1
∫ exp⎜⎜⎝ 2 ⎟⎟⎠ dt
2π − ∞
(33)
ln d − ln d 50
σ ln
(34)
1 d 84
ln
2 d 16
(35)
and
σ ln =
σ ln
d 16
d 84
μ ln,3 =
standard deviation of the distribution function
diameter at 16%
diameter at 84%
ln d 50,3
median particle size
12
-
Representation of the mass distribution Q 3 (d ) in a RRSB diagram (RRSB distribution)
and determination of the characterising parameters
⎡ ⎛ d ⎞n⎤
⎟ ⎥
Q 3 (d ) = 1 − exp ⎢− ⎜
(36)
⎢⎣ ⎝ d 63 ⎠ ⎥⎦
d 63
n
-
diameter at 63%
characterising parameter
Representation of the mass distribution Q 3 (d ) in a double logarithmic diagram and
determination of the characterising parameters (GGS distribution)
k
⎛ d ⎞
⎟ (37)
Q 3 (d ) = 0,8⎜
⎝ d 80 ⎠
k
d 80
diameter at Q3(d) = 0,8
4. 2. Determination of surface area
Please use the special instruction for the Blaine test equipment.
4. 3.Discussion of the results
- Can you describe your particle distribution with a known analytical distribution function?
- Please discuss possible error and deviations.
- What can you say about the precision of the sample splitting?
- Please discuss possible causes for occurring differences in the test sieving results of the two
analysed samples
- Evaluate with the help of the standard deviations you received for the Blaine surface area
the
reproducibility of the measurement.
5. Notes for preparation to the practical training
Basic knowledge contains of this practical training instruction as well as the lecture Mechanical process engineering.
With the help of following notes you can check your knowledge and complete. Contents of
the preliminary test for the practical training are the experiment carrying out, methods for
analysing the particle size and their application fields as well as the possibilities for the representation of particle size distributions.
13
Notes for helping the preliminary test for the practical training
Particle size, particle size distribution function, particle size frequency function, equivalent
diameter, kinds of quantities, moment, transformation of quantities, LNVT, RRSB distribution,
parameters of particle size distributions, specific surface area, SAUTER diameter, sample
splitting, median diameter, modal diameter, weighted means, deviation variance, GGS distribution, Blaine test
6.
Symbol index
A S, m
mass related specific surface area
A S, V
volume related specific surface area
D i = Q 3,i
undersize distribution
dh
d m, i
hydraulic diameter
mean class diameter
d16
d 63
d84
d ST
Δd i
Eu
Kn
k
M1,3
M
m ges
diameter at 16%
diameter at 63%
diameter at 84%
SAUTER diameter
class width
Euler number
Knudsen number
1. initial moment of the basis mass (r = 3)
index for "manometer....."
overall mass of the particulate material after sieving
m0
q 3,i
overall mass of the particulate material before sieving
frequency function
Re
T
t
u
u
ε
η
λg
Reynolds number
⎧0 = number,1 = length,2 = surfacearea ⎫
quantities
⎬
⎨
⎭
⎩3 = volume( mass)
temperature
time
running upper limit
local velocity of current
porosity
viscosity of fluids
average free path length
μ
loss mass fraction
μ 3,i
mass fraction
r
loss
μ ln,3 = ln d 50,3
ρS
ρF
σln
ψA
n
median value (expectation value)
solid density
fluid of fluids
standard deviation of the distribution function
shape factor of particles (set mostly =1)
14
7.
References
Schubert, H.:
Aufbereitung mineralischer Rohstoffe
Band 1, 4. Auflage
S. 21-43
VEB Deutscher Verlag für Grundstoffindustrie, Leipzig 1989
Schubert, H.:
Mechanische Verfahrenstechnik
Reihe: Verfahrenstechnik
3. Auflage
S. 22-45
Deutscher Verlag für Grundstoffindustrie, Leipzig 1990
K.Gotoh, H. Masuda, K. Higashitani: Powder Technology Handbook,
Marcel Dekker New York 1997

Granulometry - Lehrstuhl für Mechanische Verfahrenstechnik

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