1.0
0.8
Z = 10
1.0
0.6
θ
0.1
0.4
0.01
0.2
0.001
Similarity
Solution
0.0001
Figure 8. Nonsimilar temperature profiles
in a vertical rectangular fin. Adapted from
0
1
2
3
4
ξ
Patankar and Sparrow (1970).
1.0
Z = 10
1.0
0.8
0.1
0.01
0.6
1/
0.001
4
∆*
0.4
0.0001
Similarity
0.2
Solution
Figure 9. Similar and nonsimilar
distributions of condensate film
thickness. Adapted from Patankar
5
6
0
1
2
3
4
ξ
and Sparrow (1970).
The similarity solutions for θ, ∆*, and η are shown in Figures 8, 9, and 10, respectively.
These figures also show the nonsimilarity solutions that are discussed next.
Nonsimilarity solutions
The applicability of the similarity solutions is limited to those z locations for which the
boundary condition eq 42 is justified, that is, locations where the tip temperature is nearly
equal to the vapor saturation temperature (θ = 0). For z locations where this condition is
not met, eq 3841 were solved numerically by Patankar and Sparrow (1979). These results
appear in Figures 810.
Figure 8 shows the temperature profiles at various Z locations. For low values of Z, the
temperature distribution in the fin is quite steep as envisioned earlier. As Z increases, the
profiles become less and less steep, since larger values of θ = (Tf - Tsat )/(Tfb - Tsat ) mean
lower values for T (note that Tb Tsat is negative); one concludes from Figure 8 that the
general level of fin temperature decreases as Z increases. This confirms the earlier hypoth-
esis about the dependence of T on Z. Since the nonsimilarity temperature profiles must
terminate at X = 1, the terminal point on ξ = X / Z1/8 scale occurs at lower and lower
values of ξ as Z increases. The similarity solution, on the other hand, extends up to
ξ = 42 = 6.48 .
15
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