Page 144 - Book Hosokawa Nanoparticle Technology Handbook

FUNDAMENTALS CH. 3 CHARACTERISTICS AND BEHAVIOR OF NANOPARTICLES AND ITS DISPERSION SYSTEMS
particle motion, the equation of motion can be sim- diameter is less than 1 m. For example, equation
plified as follows [2]: (3.2.7) describing the resistance F in the Stokes
r
regime becomes the following equation:
2
dv ⎛ v ⎞
D ⎜
m p CA fr ⎟ F e (3.2.2) 3 Dv
dt ⎝ 2 ⎠ F pr (3.2.9)
r
C
c
Coefficient C D in equations (3.2.1) and (3.2.2) is
called as the resistance coefficient. The coefficient C D (2) Cunningham’s correction factor
is expressed as the function of the particle Reynolds When particle diameter D is in the same order of the
p
number defined by the relative velocity v as a repre- mean free path of gas molecule ( is about
r
sentative velocity and the particle diameter D as a 0.0645 m at 1 atm, 20°C) and/or less than that, the
p
representative length.
gas cannot be considered as a continuous medium.
Because of the molecular motion, the gas velocity at
24
C : Re p 2 (Stokes regime) (3.2.3) the surface of the particle cannot be assumed to be
D
Re p zero. In other words, the fluid slips at the surface. The
decrease of the fluid resistance due to the slip is cor-
rected by the Cunningham’s correction factor or slip
10
C 2 Re p 500 (Allen regime) (3.2.4) corrections factor.
D
Re p On the basis of the theoretical analysis or experi-
mental data, several equations giving the
Cunningham’s correction factor have been proposed.
.
C 044 : Re p 500 (Newton regime) (3.2.5)
D
One of the equations is as follows [3]:
Here, C kK n (3.2.10)
1
c
vD p 0 400exp( K )
fr
.
.
Re (3.2.6) k 1 257 . 1 10 n (3.2.11)
p

Here, K is Knudsen number defined by the ratio of
n
Substituting these equations into the first term on the mean free path to particle diameter.
right-hand side of equations (3.2.1) and (3.2.2) gives
the equation of fluid resistance force F . For example, K 2 D (3.2.12)
r
the force F in Stokes regime is as follows: n p
r
F 3 D v (3.2.7) The mean free path of the air is expressed by the
p r
r
following equation:
Equations (3.2.3) and (3.2.7) are derived analytically ⎛ M ⎞ 12
/
⎜
from the Navier–Stakes equation under the assump- 0 499P RT ⎠ ⎟ (3.2.13)
⎝
tion of without inertia. Therefore, the equation can be .
applied to the range of small Reynolds number. As for
a nano-size particle, most cases are in the Stokes , viscosity;
regime because of the small particle size. P, pressure;
Several equations describing the resistance coeffi- M, molecular weight;
cient C have been proposed besides of equations R, gas constant;
D
(3.2.3–3.2.5). For Reynolds number Re larger than T, temperature.
p
the Stokes regime, Oseen proposed the following
equation with considering the inertia effect of fluid (3) Characteristic value of particle motion
surrounding the particle [2], As one of the simplest case of particle motion, we
consider the case where a spherical particle having
24 ⎛ 3 ⎞ diameter D is injected into a stationary fluid with
p
C Re ⎝ ⎜ 1 16 Re p⎟ ⎠ : Re p 5 (3.2.8) the initial velocity v . When no external force F and
D
0
e
p
the Stokes law are assumed, the equation of motion
of the particle can be written as follows:
Here, it should be noted that the resistance coefficient
C should be corrected by the Cunningham’s correc- D 3 dv 3 Dv (3.2.14)
p
D
tion factor or slip correction factor when a particle 6 p p dt C
c
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