Che5700 陶瓷粉末處理
Batching and Mixing

Batch feed always involves mixing, objective:
- high uniformity, high reliability (more
judgment (experience) rather science)

Before mixing: action of feeding; next few
graphs showing some common equipments,
goal: good powder flowability.

Most ideal state of mixing: random
homogeneous mixture RHM
Che5700 陶瓷粉末處理
Bulk Solid Transport


Mass flow – first in, first out, good results;
Funnel flow – first in, last out (rat holing); not
desirable
One way to avoid
funnel flow: reduce
friction from walls
Che5700 陶瓷粉末處理
Properties of Mass of Particles
Pressure is not the same in all directions: one applied
pressure will create some pressures in other directions, but
always smaller; related to particle shape and packing; define
K’ = normal pressure/applied pressure
Shear applied at surface will be transmitted through a
static mass of particles
Density of mass will vary
Before flow, mass of particles will increase its volume first
(dilation)
When angular solids are piled up on a flat surface, there
will be an angle of repose; (free flowing solids: this angle is
between 15 and 30o)
Angle of Friction
Angle of storage tank and
angle of friction of particle
 whether particle can free
flow, i.e. mass flow or
funnel flow;
Arching: state when particle
can not flow at all
Common experience: flow
of powder decrease if size
of flow unit > 15% of
opening size
Che5700 陶瓷粉末處理
Cohesive & Noncohesive Solids
Non-cohesive solids (free flowing): K’ 0.35 – 0.6
Cohesiveness: often sensitive to moisture
Angle of friction: influenced by
particle size, shape, or even
water content, it often increase
cohesive force between
particles  increase angle of
friction  more difficult to free
flow
Abrasion: another
possible problem with
ceramic particles
during transportation
Che5700 陶瓷粉末處理
Powder Mixing




More art than science;
Can never achieve perfect mixing like that in fluid
phase;
Complete mixing: often refers to specific structure,
not attainable from a random process;
 Characterization of mixture: I.e. degree of
homogeneity - (1) a statistical problem; (2) sample
size (scale of scrutiny) consideration – need to be
“proper”, too large or too small: little value; e.g.
sintering after mixing – then consider diffusion
distance during sintering, choose appropriate size
for sampling，can be considered as single sample
within that size.
Scale of Segregation
•The length, area or volume of the largest region
of each component in the mixture is referred to
“scale of segregation” of that component
• In a liquid solution: minimum scale of
segregation – size of largest molecules
• In a particle system: largest particle size
Sampling size: e.g. adding
carotene into powder milk
(1/1000), how should we
sampling?
• Completely random vs
completely dispersed
• Degree of segregation
larger in the former case
Che5700 陶瓷粉末處理
Degree of Mixedness



Use statistical numbers as index of degree of mixing, to
discuss uniformity of sample, to compare different
mixing equipment and operation conditions.
Statistically, bimodal distribution for mixtures, often use
Gaussian or Poisson distribution as examples
E.g. A, B equivalent , except color different, then 
n!
r nr
p( x) 
pq
(n  r )! r!
x r/n
dp( x)
1
( x  p)
F ( x) 

exp( 
)
2
dx
2
 2
2
p,q true
fractions; x
measured
fraction
Che5700 陶瓷粉末處理
Mixing Indices
* Sampling analysis: Standard deviation s (s2 = variance) ;
where o = standard deviation of original segregated
mixture; r = standard deviation of ultimate completely
random mixture; s = standard deviation of current
sample; N = number of analyzed samples; n = particle
number in sample
N
s [
2
(
Ci

C
)

0
N 1
]1/ 2
log  o  log s
M
log  o  log  r
C1 (1  C1 ) 1/ 2
r  (
)
n
 o  [C1 (1  C1 )]
1/ 2
Che5700 陶瓷粉末處理
More Indices
For example:
Rose – M = 1 – s/o; (unmixed 0  mixed 1- 1/n1/2)
Lacey – M = (o2 – s2)/(o2 - r2) (from 0  1)
Kramer – M = (o – s)/(o - r) (from 0  1)
Hixon-Tenney-Harvey index:
if x > p  D1 = (1-x)/(1-p)
x < p  D2 = x/p
x = p  D3 = 1
 average degree of mixing Da = (N1 D1 + N2 D2 + N3
D3)/N
( x = value of some component in sample, p= expected
value)
C2 (f w )1  C1 (f w ) 2 1/ 2
 r  [C1C2 (
)]
Ms
•For different material and different size, use this
equnation to calculate RHM的variance;
•C1, C2 = fractional concentration of each component
•Ms = mass of sample
•fw  sum of product of the weight fraction f of particles
in each size class and the mean particle weight W in the
class (could it be f * w; not fw)
Che5700 陶瓷粉末處理
Mixing Analysis




Previous mixing index to evaluate mixing process or
effect of parameters, try to minimize error in sampling
and analysis.
Macro-scale mixing: by chemical analysis, phase
analysis, etc.; Micro-scale mixing: observation by
microscopy technique
E.g. use M index: to study time effect, determine optimal
condition; indicating de-mixing behavior; mixing – by
relative movement of particles: convection, shear,
diffusion (three mechanisms). Different equipment
provide different mechanisms.
Inverse of mixing  segregation (percolation of fines,
trajectory segregation, rise of coarse upon vibration)
Che5700 陶瓷粉末處理
Important Parameters
Mixing effect affected by:
 type of equipment
 energy input
 flowability and composition of sampe (size, shape,
density, surface characteristics)

For complete description of mixing, one need:
 sample variance (intensity of segregation)
 scale of segregation (to micro-scale)
 long range structure

Microscale and Macroscale
Mixedness
• Microscale analysis  provide information on
microscale mixedness
• Macroscale mixedness  can be analyzed by
many technqiues (chemical or physical)
• Sampling and analysis error should be kept to a
minimum
• uncertainty in the standard deviation (s) become
low when a large number of samples are taken
Taken from JS Reed, 2nd ed.
• Taken from JS Reed, 1995
• Commercial mixers: usually with two or more mixing
elements to produce: high shear mixing in a local region
& low shear bulk mixing
One example of two
mixing elements
To avoid vortex, we
may add baffles
Turbulence and
cavitation – for
diffusion (micro-scale
mixing)
Che5700 陶瓷粉末處理
Mechanism in Horizontal Drum Mixer
•Path of circulation: particle move with rotating
cylinder, mixing only if change in path of particles
•Radial mixing: due to mixing in the gravity direction
(drop to a void); velocity gradient important;
•Radial de-mixing: core formation, small and heavy
particles gradually go to bottom
•Axial mixing: diffusion mode
•Axial de-mixing: band formation, effect from mixer
walls;
Change drum mixer into cone shape, beneficial to
mixing; (taken from JS Reed, 1995) cement mixer!
Che5700 陶瓷粉末處理
Rate Process
•dM/dt = A (1-M) – B  (where  is segregation
potential; M = 1 - 2)  unmixing process
In principle: B –
effect from
equipment;  effect from
particle
characteristics
Sigma
blade mixer
Equipment used in mixing of viscous paste (Taken from JS
Reed, 1995); (a) helical mixer; (c) double planetary mixer
取自JS Reed, 1995
Extent of
reaction
depend on
degree of
mixing and
uniformity
Che5700 陶瓷粉末處理
Aeration Blending
Or named fluidized blending, for mixing of different
particles. Some advantages:
 can obtain uniform mixtures, even different in
density
 can be precisely controlled, energy cost/unit weight
sample lower
 easy to operate and maintain
 large capacity
 fast and easy loading; many different methods for
feeding (e.g. pneumatic, mechanical, gravity)
 Can use other gases, in addition to air

Che5700 陶瓷粉末處理
Dispersion of Powder into Liquid
Work involved with each
step (equilibrium
consideration):
Wa = SL – (LV + SV) = -LV
(cos +1) negative for any 
 Wi = 4SL - 4LV =-4LV cos
  < 90o, negative value
 Steps involved: adhesion (a Ws = (SL + LV) - SV = -LV
 b), immersion (b  c) and
(cos -1)  positive value,
spreading (c  d);
at=0o, this number is zero
In summary: work must be
done to obtain spreading
Che5700 陶瓷粉末處理
Contact Angle
cos  = (SV - SL) /LV
Wetting implies contact angle  < 90o
Che5700 陶瓷粉末處理
Macroscale Mixing and Microscale Mixing
Mixing of viscous slurry with a single impeller: difficult to
achieve both
** macrocsale mixing  high pumping capacity
** microscale mixing  turbulence
Look at two parameters, Re & P (power requirement)
** Re = N s (dia.)2/ …. turbulence
** P/Re = CD  N2 (dia.)3 …. Pumping capacity
 CD: drag coefficient;
 For high viscosity, same power, but need large Re
(microscale mixing)  need small size propeller, or increase
velocity; yet for small propeller, non-uniformity occur
(macroscale mixing)
Mixing and EMI Performance
* ABS + Ni powder or fiber  mixing
(Barbender mixer or dry mixing) composite
for EMI measurements
60
Average EMI SE (dB)
50
40
30
20
Brabender/powder
Dry mixing/powder
Dry mixing/filament
10
0
0
5
10
15
20
Nickel vol. %
25
30
Dry mixing
produced better
shielding effect
than Barbender
mixer (in terms
of low threshold
value)
Barbender mixer produced perfect mixing, not
necessary good for EMI purposes; dry mixing
produced macro-scale uniformity, not micro-scale
uniformity
(a) Barbender mixer (powder; 20%); (b) dry mixing
(powder 7%); (c) dry mixing (filament, 7%)
Correlation between percent measureable
(electrical resistance) of composite of various
samples
100
80
PM (%)
60
40
Brabender/powder
Dry mixing/powder
Dry mixing/filament
20
0
0
5
10
15
20
Nickel vol. %
25
30
Electrical
resistance
measurement
by 4 point probe