bedo of the ice is close to, but greater than zero, and
the resistive heating flux and because assuming no
depends on how much air and/or snow is incor-
resistive heating is both reasonable and conserva-
porated in the accretion, b) the shorter-wavelength
tive, it is not included in the model.
visible radiation may not be absorbed in the ice,
but it is probably absorbed in the underlying wire,
Latent heat of fusion
To freeze water at 0C, the heat of fusion must
which is assumed to be in thermal equilibrium
with the ice accretion, and c) the ground surface
be removed. The fraction f of the impinging pre-
reflects some incoming radiation, depending on
cipitation that freezes is the ratio of the sum of all
the ground cover and the amount and condition of
the other heat fluxes to the latent heat flux Qf that
snow and ice on the ground.
is calculated assuming that all the incoming water
The shortwave and longwave fluxes are oppo-
freezes:
site in sign and comparable in size, but there is no
Qc + Qe + Ql + Qs + Qw
f=
shortwave flux at night, when the air temperature
.
(21)
Qf
may be relatively cold and the longwave flux
relatively large. Both are small compared with the
If f is 1 or greater, then all the incoming water
convective and evaporative fluxes in typical freez-
freezes. Properly, the temperature of the accretion
ing-rain conditions, but large compared with the
surface should be determined iteratively, and heat-
viscous and kinetic heat fluxes (Fig. 3).
flux terms for cooling the ice to the calculated
surface temperature and depositing hoarfrost or
sublimating accreted ice should be included. These
Viscous heating
The formula for the viscous heat flux is from
are small effects in freezing-rain conditions, how-
Makkonen (1984). At wind speeds appropriate for
ever, and are not incorporated in the heat-balance
freezing rain, viscous heating from air moving by
model.
the wireice accretion is small compared with the
other heat fluxes (Fig. 3). This term is not included
Other effects
in the heat-balance model.
Two other, possibly significant, factors in de-
termining the heat balance at the ice accretion
surface are a) the angle of the wire to the wind
Kinetic heating
When precipitation drops collide with the wire
direction, and b) freezing of the impinging pre-
cipitation by the cold stored in the wire.
ice accretion, some of their kinetic energy is con-
Wind direction is archived with the weather
verted to heat. At the droplet speeds associated
data, so for a particular structure the yaw of the
with freezing rain, this term is small (Fig. 3). It is
wire to the wind direction during freezing rain
not included in the heat-balance model.
could be determined. The wire orientation could
then be incorporated in the calculation of the pre-
Resistive heating
cipitation flux w across the wire by multiplying the
If the wire conducts electricity, heat is gener-
righthand side of eq 3 by the sine of the angle
ated from the wire resistance. The resistive heat
generated per unit length of wire I2Rc depends on
between the wire and the direction of the precipi-
tation flux. The Nusselt number also depends on
the yaw of the wire. Morgan (1973, 1975) discusses
which generally is a function of the current load
the variation of the Nusselt number with yaw in
and frequency. The resistive heat flux per wire
diameter is
forced convection for both smooth cylinders and
bare, stranded wires.
102 I 2Rc
2
Qr (W/m ) =
.
(20)
If the air temperature is cold before freezing
Dc
rain begins and the wire is in equilibrium with the
The resistive heat flux in a wire varies from
air, impinging precipitation may freeze by con-
zero, when there is no current, up to values larger
ductive heat loss into the wire. The mass of water
than any of the other heating or cooling fluxes. For
that can be frozen by this mechanism is
example, for a 2.63-cm (1.036-in.)-diameter ACSR
conductor with I = 400 amp and Rc = 8.14 105
102 πDc ρccpcT
2
ohm/m (0.131 ohms/mile), Qr = 490 W/m2. This
M=
,
(22)
4Lf
is comparable to the typical convective and evapo-
where ρc is the density and cpc is the heat capacity
rative cooling fluxes in freezing rain shown in
Figure 3. Because of the extremely large range in
of the wire. This effect is small except perhaps for
11
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