k (T - Tfb ) kf w
4
kw
Z = l l sat
∆ = f 2 δ
(37)
z,
4gρl (ρl - ρv ) hfg klL2
2klL
which then become
2θ
θ
=
(38)
∆
X2
(∆4 )
=θ .
(39)
Z
The boundary conditions on θ and ∆ are
θ
X = 0, θ = 1; X = 1,
=0
(40a,b)
X
Z=0, ∆=0.
(41)
Patankar and Sparrow (1979) sought a similarity solution of eq 3841 by arguing as
follows. The Nusselt's theory on a vertical isothermal shows that the local heat transfer
coefficient hZ is proportional to Z1/4, giving high values for hZ at small values of Z. The
high values of hZ cause the fin temperature to increase rapidly from Tb at x = 0 to Tsat,
significantly before x = L. Thus the behavior of the fin closely approximates that of an
infinitely long fin, permitting the condition eq 40b to be replaced by
X=∞, θ=0.
(42)
In the limit Z = 0, hZ becomes infinite and the temperature distribution in the fin takes the
form of a step increase from Tb to Tsat. Mathematically, this means
Z=0, X>0, θ=0.
(43)
Examining the behavior of δ, one notes that at small values of Z, δ must diminish quite
rapidly with X to reflect the rapid decrease of Tsat T with X. This permits us to write
X=∞, ∆=0.
(44)
The behavior of θ and ∆ at Z = 0 and X = ∞ indicates the possibility of a similarity
solution.
Similarity solutions
The introduction of a similarity variable ξ as
ξ = X / Z1/8
(45)
with dependent variables
θ* (ξ) = θ, ∆* (ξ) = ∆4 / Z
(46)
reduces the partial differential eq 38 and 39 to the following ordinary differential equa-
tions
13
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