NuDc = 0.85 RaDc0.188
100 < RaDc < 104 (17)
(1980, p. 413) give
NuDc = 0.48 RaDc0.250
104 < RaDc < 107.
1.94
T + 273.15
κm = 0.211
1013 cm2 /s, (14)
273.15
Pa
In a separate paper, Morgan (1973) argues that
for relatively low Rayleigh numbers (100 < Ra <
which results in κm = 0.21 cm2/s at Pa = 1000 mbar
4 105), the roughness of the surface does not have
and T = 0C. They also say that conventionally
an effect on heat transfer, so these formulas should
used values may be too high, and as they are a
also apply to rough ice accretions. For most rea-
better source for this information than either
sonable combinations of the wireice accretion
Incropera and DeWitt (1985) or Batchelor (1970),
diameter and weather parameters in freezing rain,
their formula for κm is used in the heat-balance
the Rayleigh number is within this low range, so
model.
eq 17 are incorporated in the heat-balance model
The mass of water that evaporates in the pro-
to determine the heat transfer coefficient for the
cess of freezing the impinging rain is me =
wire when there is no wind.
102QeDcdt/Le (eq 9). In the heat-balance model,
The heat and mass transfer analogy can be used
the evaporated water is subtracted from the run-
in free convection (Incropera and DeWitt 1985) to
off water available for icicle formation if not
estimate the mass transfer coefficient for evapora-
all the impinging precipitation freezes directly to
tive cooling from eq 17:
the wire.
ShDc = 0.85 RaDc0.188 (Sc/Pr)0.188
100 < RaDc < 104
Convective and evaporative
ShDc = 0.48 RaDc0.250 (Sc/Pr)0.250
(18)
cooling in no wind
104 < RaDc < 107.
When there is no wind there cannot be forced
convection, so convective heat transfer occurs by
free convection only. Free convection occurs when
Longwave radiative cooling
the temperature difference between the wireice
In computing the flux of longwave radiation for
accretion and the air causes the air column to be
the heat-balance model, I assume that the wireice
accretion is radiating at 0C and the clouds and
unstable. The wireice accretion is typically warmer
than the ambient air, so the warmer air over the
surrounding rain-filled air are radiating at the air
wire rises and is replaced continuously by colder
temperature T. The emissivity of everything is
air. Heat transfer in free convection is governed by
taken to be 1 [emissivity of ice is 0.98 and of water
the Grashof number,
is 0.96 (Incropera and DeWitt 1985)]. Then the
longwave cooling flux is given by the blackbody
g(0 - T)L3
GrL =
,
(15)
radiation law:
2
(273.15 + T)νa
Ql = πσ[273.154 (T+273.15)4] .
(19)
and the Rayleigh number,
g(0 - T)L3ρacp
Shortwave radiative heating
RaL = GrL Pr =
,
(16)
The shortwave heat flux Qs = πSr/2 is deter-
(273.15 + T)νaka
mined using the measured or modeled incoming
global diffuse radiation flux Sr. This information is
specific heat of air, and νa is the kinematic viscosity
available hourly at some weather stations (NOAA
of air. It can be assumed that forced and free
1993) and is reported as an average for each hour.
The factor of π/2 transfers the reported values for
convection do not occur simultaneously when
ReL/GrL2 is much different from 1 (Incropera and
a flat horizontal surface to a horizontal cylinder.
DeWitt 1985). For typical wireice diameters, wind
The derivation of this factor is presented in Ap-
speeds, and air temperatures in freezing rain, Gr
pendix B. I assume in the heat-balance model that
<< Re2, so this is a good assumption. I assume in
the albedo of the wireice accretion is zero (the ice
the heat-balance model that forced convection
is clear), that all the incoming radiation to the
controls the heat transfer if there is any wind, and
accretion is absorbed in either the ice or the under-
free convection controls only when there is no
lying wire, and that there is no radiation reflected
wind (V = 0). The Nusselt number determined by
back from the ground to the wire. These assump-
Morgan (1975) for free convection around smooth
tions are only approximately correct, but they are
horizontal cylinders is
somewhat offsetting in their simplicity: a) the al-
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