Aerosol and Applications
First Midterm Examination
Soultions
11/10/2009
1. Terminology
a. Feret’s diameter: the maximum distance from edge to edge for the projection of particle;
b. Knudsen number: Kn =/rp, where  is the mean free path of air and rp is the characteristic
length, eg particle radius;
c. fine particle: particles with aerodynamic diameter less than 2.5 m;
d. probability density function: the derivative of cumulative distribution function, which is the
fraction of some property for each particle size, to stand for the normalized frequency or
probability;
e. aerosol: condensed matter suspended in air, generally with size from nm to mm.
2. The following results were obtained from the size distribution measurements by MOUDI:
50% size-cut (μm)
18
10
5.6
3.2
1.8
1.0 0.54 0.33 0.18 0.10 0.056
concentration (μg/m3) 0.82 1.06 1.12 1.17 1.03 1.37 2.0 2.03 1.76 1.23 1.15
a. Please find the mass median diameter and mass mean diameter; (10 points)
b.Please find the number mean diameter if assume all particles were spherical and specific density of
one. Assume the upper size-cut for the first stage is 40μm. (10 points)
c. Are the particle of log-normal distribution ? Why ? (10 points)
Solution:
50% size-cut Wi
size range
dpi wt. fraction
Fi
18
0.82
18~40
26.8
0.056
>18
0.056
10
1.06
10~18
13.4
0.072
>10
0.128
5.6
1.12 5.6~10
7.48
0.076
>5.6
0.204
3.2
1.17 3.2~5.6
4.23
0.079
>3.2
0.283
1.8
1.03 1.8~3.2
2.40
0.070
>1.8
0.353
1
0.54
0.33
0.18
0.1
0.056
1.37
2.00
2.03
1.76
1.23
1.15
14.74
1.0~1.8
1.34
0.54~1.0
0.735
0.33~0.54 0.422
0.18~0.33 0.244
0.10~0.18 0.134
0.056~0.10 0.0748
0.093
0.136
0.138
0.119
0.083
0.078
3.859
>1.0
>0.54
>0.33
>0.18
>0.10
>0.056
Ni
1.01E+01
1.05E+02
6.38E+02
3.68E+03
1.78E+04
Nf
1.24E-08
1.29E-07
7.84E-07
4.52E-06
2.19E-05
0.446
1.35E+05
0.581
1.20E+06
0.719
6.44E+06
0.839
2.90E+07
0.922
1.22E+08
1.000
6.55E+08
mass median diameter
0.782
8.14E+08
1.66E-04
1.48E-03
7.91E-03
3.57E-02
1.49E-01
8.05E-01
0.094
mass mean diameter
# mean diameter
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.01
0.1
1
10
100
The answer is no.
3. The particle size distribution in the ambient atmosphere can be grouped into five different types as
discussed by Whitby (1980) and John et al. (1992). Please draw figures for the particle size
distribution in terms of dN/d log dp and dM/d log dp, respectively and where N and M are the
number and mass concentrations, respectively. (10 points)
Solution:
4. A certain chemical species is adsorbed by particles and the mass absorbed is proportional to the
surface area of the particle. Derive an expression for the mass distribution function of the
chemical species with respect to particle size, expressed as mass of the chemical species per unit
volume of gas in the size range of dp to dp + d dp, if the number distribution function for particle
is n(dp). Define any constants you introduce. (15 points)
Solution: If all particles are assumed to be spheres and are not porous, the total surface area for the
particles is St   4 d p n(d p )d (d p ) .
2
If the chemical species is only absorbed onto the outer
surface of particle, the total mass absorbed on particles is kSt, where k is a constant.
Therefore,
the probability density function for the mass distribution of the chemical species is
4 d p n(d p )
2
f m (d p ) 
St
5. What is the aerodynamic diameter for a spherical particle with specific density of 4 and Stoke’s
diameter ofm? What is the answer if the particle is not sphere? (10 points)
Solution: Both Stoke’s diameter and aerodynamic diameter are defined to have the same terminal
settling velocity. Thus,
 p d p,s 2
d p ,a 2
gCc 
gCc , where p is the particle density, dp,s is the
18
18
Stoke’s diameter, dp,a is the aerodynamic diameter,  is the air dynamic viscosity, and Cc is the
Cummingham’s slip correction factor. If assume Cc is one, the dp,a should be 8 m. For
aerodynamic diameter of 8 m, the corresponding Cc is equal to one. Therefore, the assumption
is all right.
The definitions of Stoke’s diameter and aerodynamic diameter have nothing to do with particle
shape. Therefore, the aerodynamic diameter is still the same as 8 m.
6. The particle concentrations of different sizes emitted from a stack are shown as followings:
Particle size，m
Conc.，mg/Nm3
0.40
32
0.65
115
1.1
37
2.0
55
4.0
92
7.4
155
15
110
19
54
What is the required control efficiency for particles of 1 m, if the emission standard is 65
mg/Nm3, if the control efficiency is proportional to the particle size. (10 points)
Solution: If the control efficiency is proportional to the particle size, the control efficiency, ,
is   kd p , where dp is the particle size and k is a constant.
Thus,
(1  0.4k )32  (1  0.65k )115  (1  1.1k )37  (1  2.0k )55  (1  4.0 k )92  (1  7.4k )155
(1  15k )110  (1  19k )54  65
Therefore, k  0.132 , that is, the required control efficiency for particles of 1 m is 13.2%.
7. For the ambient secondary particles, what are the four gaseous precursors ? Please list two
possible emission sources for each precursor and state the formation pathways for each precursor
to form particular matter. (10 points)
Solution:
NOx
NH3
Airborne Particles
hv
O3
N(V), S(VI), SOC
Ox
ida
nts
H
2O
2 ,O
3 ,e
tc.
VOCs
SO2
氣
固
相
反
應
液
相
反
應
Secondary
光
化
學
反
應
SO4=
10~30%
NO3-
3~20%
NH4+
4~15%
SOA
5~20%
POA
5~10%
Primary
Primary particles,
NaCl , dust etc.
25~60%
Thus, the four gaseous precursors are SO2, NOx, NH3, and VOCs and their emission sources are:
NOx: stationary sources: power plants, boilers, …., mobile sources: cars, motorcycle, …..
SOx: stationary sources: power plant, boilers, ….., mobile sources: diesel cars,
VOCs: stationary sources: surface coatings, solvent using processes, …, car exhaust, …
NH3: agricultural farms, wetland lands, …