Eliminating qt′ between eq 68 and 70, the equation governing δ can be written as
′
-1
δ
R
l (1 - R)
dδ
(Ta - Tf ) = 0 .
k + C C h
δ2
-
(71)
dz gρl (ρl - ρv )hfg l
fa
The simultaneous solution of eq 69 and 71 gives the fin temperature Tf(z) and condensate
film thickness δ(z).
Introducing the following dimensionless variables
θ = (Ta - Tf )/(Ta - Tfb ), ∆ = δ / L, ξ = z / L
gρl (ρl - ρv ) hfg L3
2CfCahL2 (b + w)
F1 =
, F2 = Bi =
lkl (Ta - Tb )
(72)
kf bw
kl
kf bw
1
F3 =
, F4 =
=
F2 F3 2klL (b + w)
CfCahL
into eq 69 and 71 gives
d2θ
θ
=
(73)
dξ2 F4 ( ∆ + F3R)
(1 - R)θ
d∆
=
∆2
(74)
dξ F4 ( ∆ + F3R)
The case of R = 0 represents the condensation of pure vapor on a fin, and eq 73 and 74
reduce to eq 16 and 17 of Simple Models. Note for a thin fin, w/b << 1 and F4 = kw 2kl ,
which equals F2 in eq 15. It is also interesting to note that the case of purely convecting fins
with no condensation is represented by R = 1 and ∆ = 0. The range 0 < R < 1 represents the
case of simultaneous heat and mass transfer.
Figure 14 shows typical results for θ obtained from a numerical solution of eq 73 and 74
subject to the boundary condition of constant base temperature (ξ = 0, θ = 1) and insulated
tip ξ = 1, dθ dξ = 0 . The parameters F1, F2, F3 and F4 were calculated assuming moisture
mm, w = 20 mm. The value of h was calculated using the correlation of Motwani et al.
(1985) and assuming the free-stream velocity of U∞ = 4 m/s, which is typical of air-
conditioning systems. Figure 14a illustrates the effect of dry bulb temperature Ta with Tfb
= 0C and φ = 50%. As Ta increases, θ decreases, indicating an increase in fin temperature
Tf. The increase in Tf reflects higher sensible and latent heat transfer to the fin surface. The
effect of relative humidity φ shown in Figure 14b is similar to that for a radial fin (Fig. 11).
Finally, Figure 14c shows that as the fin base temperature Tfb is reduced, the driving
potential for both heat and mass transfer is increased causing the fin temperature to
increase or θ to decrease.
A comparison of dry and wet fin heat transfer is presented in Figure 15. Figure 15a
shows the ratio of total heat transfer for wet conditions, qwt, to that for dry conditions, qd.
This ratio decreases with an increase in Biot number. The ratio of sensible heat transfer for
wet conditions, qws, to qd is plotted in Figure 15b. It can be seen that the sensible heat
transfer during condensation is appreciably reduced as Bi increases. Figures 15c,d show
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