ture profiles for the two orientations are slightly different. As in the case of the horizontal
fin, a good design value of N is on the order of 10.
Equation 6 for the fin efficiency also applies to the vertical fins. The results for the
efficiency of vertical fins are shown in Figure 3 to allow a comparison with the horizontal
fin. Figure 3 shows that vertical fins are more efficient than horizontal fins and, of the two
vertical arrangements in Figure 4, the downward pointing fin has a higher efficiency than
the upward pointing fin.
Vertical rectangular fin
The results of the foregoing section are also applicable to a vertical rectangular fin if the
definition of N is modified appropriately to represent the rectangular geometry.
Conjugate models
In the conjugate models, the heat conduction equation for the fin and the condensate
boundary layer equations are solved simultaneously. In the simple model, which has been
used by Nader (1978), Burmeister (1982) and Acharya et al. (1986), both the Tf and δ are
allowed to vary along the condensate flow direction only; that is, a one-dimensional fin
model is used with a two-dimensional condensate film. The improved model proposed by
Patankar and Sparrow (1979) considers a two-dimensional fin with a three-dimensional
condensate layer.
Simple conjugate model
Consider a vertical fin of rectangular profile as shown in Figure 6. The fin has length L,
vapor at temperature, Tsat > Tfb. The boundary conditions for the fin are those of constant
base temperature Tfb < Tsat and no heat flow through the tip. To establish the conservation
equations, consider a slice (fin and two films) of thickness dx. Equating the heat conduct-
ed through the two condensate films to the net heat conducted through the fin slice gives
d2Tf -2kl (Tsat - Tf )
=
(13)
kwδ
dx2
where w is fin thickness and δ the condensate film thickness at x. In deriving eq 13, the
temperature distribution through the condensate film has been assumed to be linear. The
film thickness δ can be related to the local temperature difference, Tf Tsat, through the
application of the momentum equation, giving
d(δ 4 ) 4 lkl (Tsat - Tf )
=
.
(14)
gρl (ρl - ρv ) hfg
dx
It is convenient to introduce the following dimensionless quantities:
θ = (Tsat - Tf )/(Tsat - Tfb ), X = x / L, ∆ = δ / L
(15)
gρl (ρl - ρv ) hfgL3
kf w
F1 =
, F2 =
lkl (Tsat - Tfb )
2klL
into eq 13 and 14 to give
7
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