large structural elements. For example, for a 3-cm-
tained from eq 23 and 24 using the heat and mass
diameter aluminum wire, with cpc = 0.88 J/g C, ρc
transfer analogy and replacing Pr with Sc both
= 2.77 g/cm3, and air temperature T = 5C, the
explicitly and in Radd (Incropera and DeWitt 1985).
mass of water that can be frozen by heat conduc-
Of the other heat-flux terms, Qw = 0 for icicles
because the runoff water is already at 0C. I as-
tion into the wire is 26 g/m, which is equivalent to
an ice thickness of 0.03 cm. The calculation of this
sume that there is no solar radiation absorbed in
additional mass of ice is included as an option in
the icicles, Qs = 0, because a) the icicle orientation
the model.
is close to vertical and thus roughly parallel to the
incoming diffuse solar radiation flux, b) much of
the longer-wavelength visible radiation is absorbed
in the cloud cover, and c) very little of whatever
5. ICICLES
incident shorter-wave radiation that gets through
Precipitation that does not freeze to the wire
the clouds will be absorbed in the small-diameter
may freeze as icicles in the process of dripping off
clear icicle.
the wire. I assume that there is no precipitation
The fraction fb of the available water that freezes
impinging directly on the icicles. At low wind
is determined as in eq 21. If fb is greater than 0.64
speeds, icicles form directly under the wire and are
then all the available protoicicle water is retained,
shielded by the wire from the falling rain. At high
and if fb is less than 0.64 the fraction of the protoicicle
wind speeds, the runoff water is blown to the lee
mass that is incorporated in the icicle is fb/0.64.
side of the wire so icicles form behind the wire,
After an icicle forms, runoff water may freeze to
shielded from the wind-driven rain. In the heat-
its sides. If there is wind, the Nusselt and Sherwood
balance model, a single icicle per meter of wire
numbers for forced convection are determined
represents the actual icicles on that meter of wire
using eq 7 and 13, based on the diameter of the
stacked end to end. If there is no existing icicle, the
icicle Di. If there is no wind, heat transfer is by free
mass of water that is not frozen or evaporated
convection. Heat transfer by free convection from
while on the wire forms a protoicicle, which is a
the icicle (vertical cylinder) is different from that
stack of partially frozen hemispherical droplets
for the wire (horizontal cylinder). The relatively
with an assumed diameter dd = 0.5 cm (Maeno et
warm air at the surface of the icicle flows upward,
al. 1994). The protoicicle freezes from the outside
drawing cooler air after it. The boundary layer
in, and when the surface water freezes the unfro-
thickness increases from the bottom to the top of
zen interior water is retained and, as more heat is
the icicle, decreasing the heat transfer. The Nusselt
removed, eventually freezes. Assuming the sur-
number for free convection for this geometry is
face layer of ice on the hemispherical droplets is 0.1
based on the length of the icicles rather than on
cm thick (Fig. 2, Maeno et al. 1994), the ratio of the
their diameter. The appropriate icicle length to use
is that for a single icicle, rather than this stack of
volume of ice to the volume of the hemisphere is
icicles. Following Makkonen and Fujii (1993), I
0.64. In the heat-balance model, all the water in the
assume 45 identical icicles per meter of wire to give
protoicicle is incorporated in the icicle if the freez-
ing fraction fb, determined from the heat balance
an average icicle length Lbar = Li/45. Because the
for the protoicicle, is greater than 0.64.
icicles are closely spaced on the wire, thus inhibit-
The convective and evaporative heat fluxes for
ing cooling, I use an empirical formula for heat
this protoicicle are determined using Nudd for
transfer from a vertical plate, rather than from a
spheres. For forced convection (Incropera and
cylinder to determine the Nusselt number
DeWitt 1985, eq 7.55),
(Incropera and DeWitt 1985, eq 9.27):
Nudd = 2 + Pr0.4(0.4 Redd1/4
0.670RaLbar1/ 4
+ 0.06 Redd2/3),
NuLbar = 0.68 +
(23)
4/9
(
)
0.492 9/16
1 + Pr
and for free convection (Incropera and DeWitt
for RaLbar < 109.
1985, eq 9.35),
(25)
0.589Radd1/ 4
Nudd = 2 +
.
(24)
The Sherwood number for evaporative cooling is
[1 + (0.469 / Pr)
]
9/16 4/9
obtained from eq 25 by replacing Pr with Sc, both
explicitly and in RaLbar. The convective and evapo-
The Sherwood number for vapor transfer is ob-
rative heat fluxes are quite insensitive to the as-
12
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