Page 61 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology En001c-14 May 7, 2001 18:25

284 Aerosols

TABLE II Characteristic Transport Properties of Aerosol Par- Thus, the fall velocity is proportional to the cross-sectional
◦
ticles of Unit Density in Air at 1 atm and 20 C a area of the particle, and the ratio of its density and the gas
Mean viscosity (for values, see Fig. 1). If the particle Reynolds
Particle thermal Mean number approaches or exceeds unity, Stokes’ theory must
radius Mobility Diffusivity speed ¯ν p free path be modified.
2
(cm) B (s/g) D p (cm /s) (cm/s) λ p (cm)
In terms of the drag coefficient C D , the drag force is
1 × 10 −3 2.94 × 10 5 1.19 × 10 −5 4.96 × 10 −3 6.11 × 10 −6 written:
5 × 10 −4 5.96 × 10 5 2.41 × 10 −8 1.41 × 10 −2 4.34 × 10 −6 2
1 × 10 −4 3.17 × 10 6 1.28 × 10 −7 0.157 2.07 × 10 −6 C D = ρ g π R U 2
2
5 × 10 −5 6.71 × 10 6 2.71 × 10 −7 0.444 1.54 × 10 −6 ∞
The results of experimental measurements for spheres in
1 × 10 −5 5.38 × 10 7 2.17 × 10 −6 4.97 1.12 × 10 −6
a fluid indicate that the drag coefficient can be expressed
5 × 10 −6 1.64 × 10 8 6.63 × 10 −6 14.9 1.20 × 10 −6
as:
1 × 10 −6 3.26 × 10 9 1.32 × 10 −4 157 2.14 × 10 −6
5 × 10 −7 1.26 × 10 10 5.09 × 10 −4 443 2.91 × 10 −6 12 2/3
C D = (1 + 0.251Re )
1 × 10 −7 3.08 × 10 11 1.25 × 10 −2 4970 6.39 × 10 −6 Re
a where the multiplier of the term outside the parentheses is
cgs units.
thedragcoefficientforStokes’flow.TheReynoldsnumber
Re ≡ U ∞ R/ν g .
If Kn is not assumed to be zero, then the Stokes drag
= 6πµ g RU ∞
forceontheparticlealsomustbecorrectedforaslippageof
where U ∞ is the gas velocity far from the particle and µ g
gas at the particle surface. Experiments of Robert Millikan
the gas viscosity.
and others showed that the Stokes drag force could be
The particle mobility B is defined as B ≡ U ∞ / . Gen-
corrected in a straightforward way. Using the theory of
erally, the particle velocity is given in terms of the prod-
motion in a rarified gas, the mobility takes the form:
uct of the mobility and a force F acting externally on
the particle, such as a force generated by an electrical B = A/6πµ g R
field. Under such conditions, the particle motion is called
“quasi-stationary.” That is, the fluid particle interactions Here, the numerator is called the Stokes–Cunningham fac-
are slow enough that the particle behaves as if it were in tor. The coefficient A is
steady motion even if it is accelerated by external forces. −1
A = 1 + 1.257Kn + 0.400Kn exp(−1.10Kn )
Mobility is an important basic particle parameter; its vari-
ation with particle size is shown in Table II along with based on experiments. Thus, the mobility increases with
other important parameters described later. increasing Kn, reflecting the increasing influence of a rar-
The analogy for transport processes is readily inter- ified gas molecular transfer regime.
preted from Stokes’ theory if we consider the generaliza-
tion that “forces” or fluxes of a property are proportional
2. Phoretic Forces
to a diffusion coefficient, the surface area of the body, and
a gradient in property being transported. In the case of mo- Particles can experience external influences induced
mentum, the transfer rate is related to the frictional and by forces other than electrical or gravitational fields.
pressure forces on the body. The diffusion coefficient in Differences in gas temperature or vapor concentration
this case is the kinematic viscosity of the gas (ν g ≡ µ g /ρ g , can induce particle motion. Electromagnetic radiation also
where ρ g is the gas density). The momentum gradient is can produce movement. Such phoretic processes were ob-
µ g U ∞ /R. served experimentally by the late nineteenth century. For
If the particles fall through a viscous medium by the in- example, in his experiments on particles, Tyndall in 1870
fluence of gravity, the drag force balances the gravitational described the clearing of dust from air surrounding hot
force, or: surfaces. This clearance mechanism is associated with the
thermal gradient established in the gas. Particles move in
4 3
3 (ρ p − ρ g )gπ R = 6πµ g Rq s the gradient under the influence of differential molecular
where g is the gravitational force per unit mass. Since bombardment on their surfaces, giving rise to the ther-
ρ p
ρ g , the settling velocity is mophoretic force. This mechanism has been used in prac-
tice to design thermal precipitators for particles. Although
2
2R ρ p g this phenomenon was observed and identified as being
q s =
proportional thermal gradients, no quantification of the
9µ g