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Particle Technology Laboratory
Prof. Sotiris E. Pratsinis
Sonneggstrasse 3, ML F13, ETH Zentrum
Tel.: +41-44-632 25 10
http://www.ptl.ethz.ch
151-0902-00 Micro- and Nano-Particle (MNP) Technology FS17
Exercise 1: Particle Size Distributions
Problem 1
For the particle size distribution data of Table 1, calculate the following average
diameters:
a) arithmetic mean, b) geometric mean, c) count median, d) diameter of average
mass, e) mass mean and f) surface mean (or Sauter).
Table 1. Particle size distribution data.
Particle diameter, nm
1
3
5
8
Number
3
5
2
1
Problem 2
Carbonaceous (soot) agglomerates emitted by combustion engines have an
asymptotic lognormal mobility size distribution with geometric standard deviation,
σg,m, of 1.8 ± 0.1, regardless of the operating conditions [1]. In natural gas engines,
the soot agglomerate mass, m, is related to the mobility diameter, dm, by [2]:
m = 0.04 d m2.45
a) If the count median mobility diameter, dm,g, is 95 nm, calculate the average soot
agglomerate mass.
b) The diesel soot agglomerates typically consist of polydisperse primary particles
with count median diameter, dp,g, of 25 nm and geometric standard deviation, σg,p,
of 1.2. Calculate the average number of primary particles per agglomerate. The
bulk soot density is 1800 kg/m3.
References:
[1] S.J. Harris, M.M. Maricq, Signature size distributions for diesel and gasoline engine exhaust particulate matter, J. Aerosol Sci. 32 (2001) 749‐764.
[2] B. Graves, J. Olfert, B. Patychuk, R. Dastanpour, S. Rogak, Characterization of Particulate Matter Morphology and Volatility from a Compression‐Ignition Natural‐Gas Direct‐ Injection Engine, Aerosol Sci. Technol. 49 (2015) 589‐598.
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Problem 3
The particle size distribution of a powder was analyzed by sieving and the following
residues were measured:
Mesh size/ m
Residue / g
0
45
63
90
125
180
250
355
500
1.1
1.9
5.25
7.5
21.8
37.5
37.5
97.0
90
710
1000 1400
112.5 285
52.5
a) What kind of distribution is shown in the table? Calculate and plot the fraction and
the cumulative fraction versus the particle diameter.
b) The above distribution can be described by a lognormal distribution. Calculate the
mass median diameter (MMD) and the σg of this distribution. Then, calculate the
CMD.
Hint: for lognormal distributions, Eq. 4.46 from Hinds (1999) can be used:
1/2
d 
d
d
 g  84%  50%   84% 
d50% d16%  d16% 
where da% is the diameter for which a% of the total “amount” of particles are
smaller and (100-a)% are larger. “Amount” can be the number, mass, area (etc.),
depending on the kind of the available distribution.
Problem 4
Titania (TiO2) particles are used as pigments for paints, in papers and in the
fabrication of plastics [3]. Figure 1 shows transmission electron microscopy (TEM)
images of TiO2 nanoparticles generated in a flame reactor and sampled at different
heights above the burner (HAB = 5, 8, 13 and 23 mm). The TEM images can be used
to determine the particle size distribution of each sample. For one of these samples
the following primary particle diameters (in nm) where measured:
17.9
22.7
20.3
13.6
13.5
27.2
15.9
15.2
22.4
25.1
13.3
15.5
27.2
28.6
9.3
16.0
19.8
22.7
12.3
7.5
21.9
11.8
19.4
22.9
22.0
14.5
26.3
24.5
18.8
24.1
a) What is the geometric mean diameter and the σg of this distribution?
b) The average primary particle diameter measured by nitrogen adsorption
corresponds to the Sauter diameter. What is the Sauter diameter determined
using the above distribution?
c) Now, you can determine on your own the particle size distributions and answer to
questions a) and b) for each HAB, using the ImageJ software (which can be
downloaded under: http://rsbweb.nih.gov/ij/.)
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Figure 1. Transmission electron microscopy image of titania nanoparticles [1].
Reference
[1] H.K. Kammler, L. Mädler, S.E. Pratsinis, Flame Synthesis of Nanoparticles,
Chem. Eng. Technol. 24 (2001) 583-596.
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