Again using the ideal gas law and eq A2, this can be rewritten as
NuD ka mwLe (e0 eT )
Qe =
.
(A9)
D ρacp R(T + 273.15)
Makkonen's formulation differs from this model's by the temperature used to
calculate the saturation vapor density at the freezing surface, by the dependence on
ka/(ρacp) rather than on [ka/(ρacp)]0.37κm0.63, and by using the saturation vapor
density of the air rather than the vapor density at the ambient humidity.
The effect of these differences is to make Lozowski and Gates's Qe 4% larger and
Makkonen's Qe 32% smaller than this model's at typical freezing-rain conditions
T = 3C, Rh = 90% and Pa = 1000 mbar. The differences between the models increase
somewhat as air temperature decreases and increase substantially as humidity
decreases.
This comparison addresses the differences in the physical models of evaporative
cooling and in the assumed dependence of the Nusselt number on the Prandtl and
Schmidt numbers. There are further differences between the models in the empirical
formulas used to relate the Nusselt number to the Reynolds number and Rayleigh
number.
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