∆ρw = ρw0 - RhρwT
Reynolds numbers to intersect Morgan's at ReD =
1600. In Morgan's formula for NuD, νa and ρa are
RheT mw ,
e0
= 10-4
supposed to be determined at the average of the
(11)
-
T + 273.15 R
T
273.15
wireice surface temperature (0C) and the air
temperature T. However, these temperatures typi-
where mw is the molecular weight of water vapor,
cally differ by only a few degrees in freezing rain,
R is the gas constant, Rh is the relative humidity,
so there is very little error in using the air tem-
and eT and e0 are the saturation vapor pressures at
perature in calculating these parameters.
temperatures T and 0C, respectively. This formu-
Both Achenbach's and Morgan's experiments
lation for Qe is compared with that in other ice
were done in air, and neither specified the depen-
accretion models in Appendix A.
dence of NuD on the Prandtl number. For air at the
The empirical formula used in the heat-balance
temperatures and pressures appropriate for freez-
model for the saturation vapor pressure over wa-
ing rain, Pr is essentially constant. Following
ter,
Zhukauskas (Incropera and DeWitt 1985), I as-
sume that NuD is proportional to Pr0.37. Thus eq 7
eT = 6.1121(1.0007 +
3.46 106Pa)e17.502T/(240.97 + T) ,
can be written
(12)
NuD = (0.583/Pr0.37) Pr0.37 ReD0.471
is from Buck (1981). The saturation vapor pressure
35 < ReD < 1600
over water, rather than over ice, is used, both in the
NuD = (0.18/Pr0.37 ) Pr0.37 ReD0.63
air and at the accretion surface, because both the
1600 < ReD < 1 106
(8)
precipitation and the accretion surface are unfro-
NuD = (0.00257/Pr0.37) Pr0.37 ReD0.98
zen.
ReD > 1 106.
The vapor transfer coefficient hm is determined
using the mass and heat transfer analogy (Incropera
This formulation is used with the heat and mass
and DeWitt 1985). The nondimensional number
transfer analogy in determining the evaporative
for mass transfer equivalent to NuL for heat trans-
fer is the Sherwood number ShL = hmL/κm where
cooling flux.
κm is the vapor diffusivity. The Schmidt number
Sc = νa/κm is equivalent to Pr. Like Pr, Sc is
Evaporative cooling
Under typical freezing-rain conditions, the im-
virtually constant for air at the temperatures and
pinging water does not freeze immediately. There
pressures we are concerned with. The heat and
is a vapor density gradient away from the wire
mass transfer analogy states that for any function
because the wireice surface is warmer than the air
NuL(ReL, Pr) the Sherwood number is the same
and the saturation vapor density increases with
function of ReL and Sc. Thus, for mass transfer the
temperature. This gradient is enhanced if the rela-
equivalent to eq 8 is
tive humidity of the air is less than 100%. Thus, the
water film on the surface of the wireice accretion
ShD = (0.583/Pr0.37) Sc0.37 ReD0.471
is evaporating as it freezes and cooling the remain-
ing water. The evaporative cooling flux is
35 < ReD < 1600
ShD = (0.18/Pr0.37) Sc0.37 ReD0.63
Leme
1600 < ReD < 1 106
Qe = 102
= 106 πLehm ∆ρw ,
(9)
(13)
Dcdt
ShD = (0.00257/Pr0.37) Sc0.37 ReD0.98
ReD > 1 106.
mass of evaporated water per unit length, Dc is the
The vapor transfer coefficient hm = (ShDκm)/D is
wireice diameter, dt is the time interval, hm is the
vapor transfer coefficient, and ∆ρw is the vapor
then obtained from eq 13 for the appropriate
density difference between the ice accretion sur-
Reynolds number range. Significantly different
face and the air. The vapor density ρwT is related to
values for vapor diffusivity κm are specified in
the saturation vapor pressure eT by the ideal gas
different references. Incropera and DeWitt (1985)
give κm = 0.26 cm2/s at T = 25C and Pa = 1000
law
mbar in their Table A8. They also say κm is propor-
eTmw
tional to (T + 273.15)3/2, which results in κm = 0.23
= 104 R(T + 273.15) ,
(10)
ρwT
cm2/s at 0C. Batchelor (1970, Appendix 1) speci-
fies κm = 0.25 cm2/s at 15C. Pruppacher and Klett
to give
9
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