Page 62 - 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

Aerosols 285

phenomenon was made until the 1920s. Einstein in 1924 When the particle is moving relative to the suspending
discussed a theory for the phenomenon; others measured fluid, transport of heat or matter is enhanced by convective
the thickness of the dust-free space in relation to other pa- diffusional processes. Under conditions where the particle
rameters. Much later, in the 1960s, the theory was refined exists in a rarified medium (Kn
0), the heat and mass
and extended. The theoretical relation for the thermal or tranfer relations are modified to account for surface ac-
thermophoretic force F t on a spherical particle is commodation or sticking of colliding molecules and the
slippage of gas around the particle.
2
F t =−KR (k g /¯υ g )
T/
R
where k g is the thermal conductivity of the gas, ¯υ g the
4. Accelerated Motion
mean thermal velocity of the gas molecules
T/
R
the temperature gradient at the particle surface, and K When particles are accelerated in a gas, their motion is
a factor depending on Kn and other particle parameters governed by the balance between inertial, viscous, and
[K ≈ (32/15) exp(−αR/α g ), where α ≈ unity but is a external forces. An important characteristic scale is the
function of momentum and thermal accommodation of time for an accelerated particle to achieve steady motion.
molecules on the particle surface]. To find this parameter, the deceleration of a particle by
Similar expressions can be derived for other phoretic friction in a stationary gas is considered. In the absence of
forces reflecting different effects of gas nonuniformities. external forces, the velocity of a particle (q) traveling in
the x direction is calculated by:
3. Heat and Mass Transfer (dq/dt) + U ∞ = 0

Particles suspended in a nonuniform gas may be subject or
to absorption or loss of heat or material by diffusional q = q 0 exp(− t)
transport. If the particle is suspended without motion in
2
a stagnant gas, heat or mass transfer to or from the body if the initial velocity is q 0 and = 9µ g /2ρ p R A. The
can be estimated from heat conduction or diffusion theory. distance traveled by the particle is, in time t,
One finds that the net rate of transfer of heat to the particle t
−1

surface in a gas is x = qdt = q 0 [1 − exp(− t)]
0
φ H = 4π RD T (T ∞ − T s )
The significance of is then clear; it is a constant that is
where ∞ and s refer to free stream and surface conditions the reciprocal of the relaxation time for stopping a particle
and D T is the thermal diffusivity k g /ρ g C p (C p is the spe- in a stagnant fluid. Similarly, on can show that 1/ repre-
cific heat of the particle). For mass transfer of species A sents the time for a particle falling in a gravitational field
through B to the sphere, to achieve its terminal speed. Note that the terminal speed
−1
q s = g .As t/ →∞, the distance over which the
φ m = 4π RD AB (ρ A∞ − ρ AS ) −1
particle penetrates, or the stopping distance L,is q 0 .
where D AB is the binary molecular diffusivity for the two
gases and ρ A∞ and ρ AS are the mass concentrations of
5. Curvilinear Particle Motion
species A far from the sphere and at the sphere surface.
This relation is basically that attributed in 1890 to Robert When particles change their direction of movement, as for
Maxwell. His equation applied to the steady-state evapo- example around bluff bodies such as cylinders or bends
ration from or condensation of vapor component A in gas in tubing, inertial forces tend to modify their flow paths
B on a sphere. Maxwell’s equation is analogous to Stokes’ relative to the suspending gas. Particles may depart from
law. the path of gas molecules (streamlines) and collide with
The applicability of Maxwell’s equation is limited in the larger body (Fig. 2). This is the principle underlying
describing particle growth or depletion by mass transfer. inertial particle collectors.
Strictly speaking, mass transfer to a small droplet cannot The trajectory of a particle moving in a gas can be esti-
be a steady process because the radius changes, causing a mated by integrating the equation of motion for a particle
change in the transfer rate. However, when the difference over a time period given by increments of the ratio of the
between vapor concentration far from the droplet and at radial distance traveled divided by the particle velocity,
the droplet surface is small, the transport rate given by that is, r/q. Interpreting the equation of motion, of course,
Maxwell’s equation holds at any instant. That is, the dif- requires knowledge of the flow field of the suspending gas;
fusional transport process proceeds as a quasi-stationary one can assume that the particle velocity equals the fluid
process. velocity at some distance r far from the collecting body.