Page 145 - Book Hosokawa Nanoparticle Technology Handbook

3.2 SINGLE NANOPARTICLE MOTION IN FLUID FUNDAMENTALS
The distance S of particle movement at time t is as Cunningham correction factor Cc is increased, for
follows: example, by increasing the mean free path by means
of the low-pressure operation. This means that per-
S { exp( t )} (3.2.15) formance of the inertia classifier such as the impactor

v 1
0
can be improved by means of the operation in low
pressure.
2
DC
p p c (3.2.16)
18
References
Index defined by equation (3.2.16) has the dimen- [1] C. Crowe, M. Sommerfeld and Y. Tsuji: Multiphase
sion of time and it is called as “particle relaxation Flows with Droplets and Particles, CRC Press, Boca
time”. The distance S at time t is called as the Raton, Florida, USA, 86 (1997).
“stopping distance”. The distance S represents the [2] Y. Otani, H. Emi: J. Soc. Powder Technol., Jpn., 23,
inertia effect on the particle movement. 31–43 (1986).
3
If the gravity mg (m: mass ( ) D /6, g: [3] P.C. Reist: Introduction to Aerosol Science, Macmillan
p
f
p
acceleration of gravity) is acting on the particle as an Pub., New York, USA, 53 (1984).
external force, the motion of equation and the particle
velocity at time t can be written as follows:
3.2.2 Phoretic phenomena
dv 3 Dv
p
D 3 p mg (3.2.17) 3.2.2.1 Phoretic phenomena in gas phase
p
p
6 dt C c The phoretic phenomenon is defined as the particle
motion caused by a force acting non-uniformly on a
v 1 { exp( t )} (3.2.18) particle surface or by the motion of a medium in the

g
very vicinity of the surface because of interactions
between the particle surface and the medium. In gen-
From equation (3.2.18), it can be found that the veloc- eral, a phoretic phenomenon is distinguished from the
ity will become terminal velocity v ( g) at particle motion caused by forces acting on the entire
t
t . The particle relaxation time is equal to the part of each particle, such as external forces.
1
time when the particle velocity v reaches (1 exp )v t The phoretic motion of particles in the gas phase is
( about 0.632v ). It characterizes how fast the parti- induced, as shown in Fig. 3.2.1, when the momentum
t
cle reaches the steady state. transferred from the gas molecules to the particle sur-
By using the characteristic length D and the char- face is not uniform over the surface. Particles sus-
acteristic velocity U specifying the system in which pended in a gas show Brownian motion due to
the particle is moving, equation (3.2.14) can be collision with the surrounding gas molecules that
rearranged into non-dimensional form, show thermal motion. For uniform temperature and
gas composition, the motion of gas molecules is
isotropic and random. In such a case, the momentum
dv
v (3.2.19) transferred to the particles is uniform over their entire
dt surface after the collisions of many molecules, result-
ing in isotropic particle movement.
C D U Sk U However, in a field with non-uniform temperature
2
cp p (3.2.20) and composition, the momentum transferred to a par-
18 D 2 D
ticle surface is dependent on the position on the sur-
face, resulting in a biased motion and the net transport
The coefficient of the inertia term ( left-hand of particles, i.e., phoretic motion. Typical phoretic
side) of equation (3.2.19) is “inertia parameter”. If the phenomena in the gas phase include thermophoresis
inertia parameter and Reynolds number Re charac- originating from a temperature gradient around a
terizing the fluid flow are the same in two different particle and diffusiophoresis originating from a non-
systems, the particle motions in these systems are uniform distribution of gas composition. The evalua-
similar. The coefficient of the inertia term obtained by tion of phoretic velocity caused by such phoretic
the characteristic length D/2 is called as Stokes num- phenomena is described below.
ber Sk 2 . The inertia parameter and Stokes
.
.
number Sk are the ratio of the stopping distance U to (1) Thermophoresis
the characteristic length D. Thermophoresis occurs owing to the difference in the
As it can be found in the definition equation momentum transfer to a particle between gas mole-
(3.2.20) of the inertia parameter, the particle shows cules with a high thermal velocity and those with a low
the similar motion with the larger particle when the thermal velocity. Because this phenomenon enhances
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