Page 171 - Instrumentation Reference Book 3E

Terminal velocity 155
, Ferh I , Martin The relationship between the terminal velocity of

I I a particle and its diameter depends on whether
I I I I I I I the flow local to the particle is laminar or turbu-
0 quickly than in turbulent flow where particles tend
lent. In laminar flow, the particle falls more
to align themselves for maximum drag and the
drag is increased by eddies in the wake of the
particles.
The general equation for the drag force F on a
particle is:

Projected Image
area I shear where K is the drag coefficient depending on
I ! shape, surface. etc., d is the particle dimension
i I I I I
(diameter of a sphere), V is relative velocity, 11 is
fluid viscosity. po is fluid density, and ?I varies
from 1 for laminar flow to 2 for turbulent flow.
For some regularly shaped particles K can be
calculated. For example, for a sphere in laminar
flow, K = 37r.
Hence from the above, we find for laminar flow
Figure 11.1 Statistical diameters.
spheres
F = 37rdVq
the mean diameter. Consider a large number of
identical particles, “truly” randomly orientated on and this is known as Stokes’s law.
a microscope slide. The mean of a given measure- By equating drag force and gravitational force
ment made on each of the particles but in the same we can show
direction (relative to the microscope) would yield a
statistical mean diameter. The following is a series
of statistical mean diameters that have been pro-
posed (see Figure 1 1.1): where p is particle density, and VT is terminal
velocity. Thus,
(a) Fergt diameter: the mean of the overall
width of a particle measured in all directions.
(b) Martin’s diameter: the mean of the length of
a chord bisecting the projected area of the If the terminal velocity of irregularly shaped par-
particle measured in all directions. ticles is measured together with p and T, the value
(c) Projected area diameter: the mean of the diam- obtained for d is the Stokes diameter. Sometimes
eters of circles having the same area as the the term “aerodynamic” diameter is used, denot-
particle viewed in all directions. ing an equivalent Stokes sphere with unit density.
(d) Image shear diameter: the mean of the dis- Stokes diameters measured with spheres are
tances that the image of a particle needs to be found to be accurate (errors < 2 percent) if the
moved so that it does not overlap the original Reynolds number
outline of the particle, measured in all direc-
tions.
In microscopy, because particles tend to lie in a
stable position on the slide, measurements as is less than 0.2. At higher values of Re; the calcu-
above of a group of particles would not be ”truly” lated diameters are too small. As Re increases, 72
randomly orientated. In these circumstances, the increases progressively to 2 for Re > 1000 when
above diameters are “two-dimensional statistical the motion is fully turbulent, and according to
mean diameters.” Newton the value of Kfor spheres reduces to d16.
For very small particles where the size
approaches the mean free path of the fluid mole-
11.3 Terminal velocity cules (-0.1 pm for dry air at s.t.p.) the drag force
is less than that predicted by Stokes. Cunningham
The terminal velocity of a particle is that velocity devised a correction for Stokes’s equation:
resulting from the action of accelerating and drag
forces. Most commonly it is the free failing speed
of a particle in still air under the action of gravity.