1.0
Non-similarity
Solution
Similarity
η
Solution
(a)
0.1
0.01
0.1
1.0
10
1.0
η
Non-similarity
Similarity
Solution
Solution
(b)
0.1
0.0001
0.001
0.01
0.1
Z
Figure 10. Comparison of similarity and nonsimilarity results for the fin
efficiency. Adapted from Patankar and Sparrow (1970).
The condensate film thickness results are shown in Figure 9. For a fixed Z location, the
film thickness decreases as x or ξ increases, and this is consistent with the temperature
differential (Tsat T) decreasing along X. For small Z, the film is highly nonuniform along
the X direction but becomes more and more uniform as Z increases. This clearly shows
that the assumption δ = δ(Z) employed in previous sections is not strictly valid.
Figure 10 shows the fin efficiency as a function of Z. The lower two curves cover the
range of Z from 0.0001 to 0.1, while the upper two curves cover the Z values from 0.01 to
10. Interestingly the similarity and nonsimilarity solutions for η are virtually identical up
to Z = 0.01. Thus for Z ≤ 0.01, eq 54 for η and hence eq 52 for q(Z) give accurate predictions.
However, this is not true of the similarity results for θ and ∆*. For example, Figure 8 and 9
shows that for Z = 0.01, there is a significant difference between the similarity and the
nonsimilarity solutions.
Example 3
A vertical rectangular fin (k = 400 W/m K) is attached to a cooled vertical surface (Fig.
7). The fin dimensions are L = 1.5 cm, w = 1.5 mm, and H = 25 cm. The environment
surrounding the fin is saturated steam at 50C. Calculate (i) the rate at which heat is
removed by the cooled surface, (ii) the condensation rate supported by the fin, (iii) the fin
temperature and the film thickness at H = 1.5 cm, z = 25 cm.
Solution
The density and heat of vaporization for steam at 50C are
ρv = 0.082 kg / m3 , hfg = 2.383 106 J/kg .
Evaluating the properties of water (condensate) at a mean temperature of (24 + 50)/2 =
37C, we have
ρl = 993 kg /m 3 , l = 694 10-6 N s/ 2 , kl = 0.628 W/m K .
m
Using the above properties in eq 37, Z can be evaluated:
k (T - Tfb ) kf w
4
Z = l l sat
z = 0.01 .
4g ρl (ρl - ρv ) hfg klL2
16
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