sections and cylinders with other diameters.
velocity of drizzle and rain drops, typically 100s of
A slightly more complex model takes the in-
microns to 1 or 2 mm in diameter, is between about
creased flux of water due to the horizontal velocity
2 and 7 m/s (Wang and Pruppacher 1977). Thus,
of the rain drops in the wind into account. Using
even in no wind, there is a downward flux of water
the flux of precipitation from eq 3, the uniform
that is a balance between gravity and the drag of
radial ice thickness is
the still air on the drops. The drops fall through the
air and collide either with the ground or with
(
)
D N
2
∑ 0.1Pjρo
Req =
+
structures in their path. When there is wind, the
Sρi j =1
(27)
drops gain horizontal velocity in addition to their
vertical velocity. The effect of wind on drizzle- and
(
)
2 1/2
2
+ 0.36WjVj
∆t .
rain-sized drops could be determined by numeri-
cally integrating the trajectory equation (Lozowski
This simple flux model with S/D = π is incorpo-
and Oleskiw 1983), including the gravitational
rated in ZRAIN for comparison with the heat-
term:
r
balance model.
C ρ r r r r
(
)
r
dv
= 3 D a v V v V + g ,
(28)
4dρo
dt
7. COLLISION EFFICIENCY
r
r
where v is the drop velocity, V is the wind veloc-
In in-cloud icing, the collision efficiency of the
ity, CD is the drag coefficient of the drops, d is the
r
cloud droplets with a structure is an important
drop diameter, and g is the acceleration of grav-
factor in determining the rate of ice accretion.
ity. In the absence of a solution to this equation for
Cloud droplets are very small, typically 5 to 50 m
rain- and drizzle-sized drops, it is reasonable to
in diameter, and have correspondingly small ter-
assume that the drops fall at their terminal veloc-
minal velocities (0.007 m/s for 15 m droplets,
ity, move horizontally at the wind speed, and
Best 1950). Thus, in windless conditions there is
collide with all obstacles in their path.
essentially no droplet flux. When there is wind, the
In some freezing-rain ice accretion models, the
wind drag on the droplets carries them along and
collision efficiency of the rain and drizzle drops is
the droplets follow the wind streamlines around
calculated incorrectly. The resultant velocity of the
any obstacle in their path. Only the inertia of the
wind speed and the drops' terminal velocity is
droplets, which tends to keep them moving in a
used in the Langmuir and Blodgett (1946) formu-
straight-line path, makes them diverge from the
lation to determine the collision efficiency. This
wind streamlines. The collision efficiency of the
ignores the different physics of falling and wind-
cloud droplets with a structure represents a bal-
carried droplets and leads to obviously ridiculous
ance between their drag, the tendency to follow
results: 0.5-mm-diameter drizzle drops falling at 2
the streamlines, and their inertia, the tendency to
m/s in no wind are calculated to have a collision
continue in a straight line. Qualitatively, the re-
efficiency of 0.56 with a 1-m-diameter cylinder
sults of this balance are that a) smaller droplets,
and 0.05 with a 10-m cylinder. In that world, if you
which have less inertia, have lower collision effi-
carried a big enough umbrella, not only would
you stay dry, so would the umbrella.
ciencies, b) the collision efficiency of droplets is
smaller for large obstacles than for small ones,
because streamlines diverge relatively farther in
front of large obstacles than small ones, and c) in
8. WIND LOAD
high winds droplets have more momentum so
It is often important to know the wind load on
their collision efficiency is higher. These observa-
a structure both during a freezing-rain storm, and
tions are quantified in Langmuir and Blodgett
for as long after the storm as ice remains on the
(1946). They numerically solved the trajectory equa-
structure. The projected area of the structure is
tion, ignoring gravitational effects, to determine
larger because of the ice accretion, so at a given
droplet collision efficiency as a function of two
wind speed the wind load is greater than it would
nondimensional numbers that depend on wind
be on the bare structure. In ZRAIN, the wind load
speed, wire diameter, droplet diameter, and the
per meter of wire is determined every hour using
density and viscosity of air.
the wireice diameter and the icicle diameter and
The Langmuir and Blodgett results cannot be
length, assuming a drag coefficient CD = 1 for both:
applied to precipitation droplets. The terminal
14
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