APPENDIX A: COMPARISON OF EVAPORATIVE
HEAT FLUX FORMULATIONS
The formulation of the evaporative heat flux term in the heat-balance model is
different from that of Lozowski et al. (1987) and Makkonen (1984). The different Qes
are compared in this appendix. Assuming consistent units and considering the heat
flux across an arbitrary surface, Qe is (eq 4)
RheT mw
e0
Qe = hmLe
,
(A1)
-
T + 273.15 R
273.15
Using the definition of the Sherwood number ShD and the relationship between ShD
and the Nusselt number NuD specified by the heat and mass transfer analogy gives
NuD = hD/ka = CPr0.37 ReDb
(A2)
ShD = hmD/κm = CSc0.37 ReDb
where C and b depend on ReD. Equation A2 can be used to write hm in terms of NuD:
0.37
κ m NuD Sc
Pr
hm =
.
(A3)
D
Using this relationship and Sc/Pr = ka/(ρacpκm), eq A1 can be written finally as
0.37
k
RheT
mwLe e0
NuD
κ m0.63
Qe =
a
.
(A4)
R 273.15 T + 273.15
ρacp
D
The evaporative heat transfer term in Lozowski et al. (1987) is given as
e - Rhe
0.63
Pr
Qe = h
εLe 0
T ,
(A5)
Sc
cp Pa
where ε = mw/ma is the ratio of the molecular weights of water vapor and air. Using
the ideal gas law,
ρaR(T + 273.15)
Pa =
,
(A6)
ma
and the definition of NuD in eq A2, eq A5 becomes
0.37
mwLe (e0 RheT )
NuD ka
κ m0.63
Qe =
.
(A7)
D ρacp
R(T + 273.15)
Thus, Lozowski and Gates's formulation (eq A7) is different from this model's
formulation (eq A4) only in the temperature used to calculate the saturation vapor
density at the accretion surface.
In Makkonen (1984) the evaporative cooling flux is given as
hεLe (e0 - eT )
Qe =
.
(A8)
cp Pa
20
Previous Page
