d2θ
θ
=
(16)
∆F2
2
dX
d(∆4 ) 4θ
=
.
(17)
dX
F1
The boundary conditions for eq 16 and 17 are
dθ
X = 0,
∆ = 0,
= 0 ; X = 1, θ = 1 .
(18a,b)
dX
Nader (1974) obtained a numerical solution of eq 1618, while Burmeister (1982) devel-
oped an approximate analytical solution of the same equations. Acharya et al. (1986)
solved the dimensional equations (13 and 14) using an iterative numerical scheme. They
also extended their computations to six other fin shapes (triangular, trapezoidal, convex
parabolic, concave parabolic, cylindrical, and conical) and developed simple correlations
for the fin efficiency.
Nader's solution
Nader (1974) introduced a new variable ψ = ∆4 which enabled him to transform eq 16
and 17 into two, coupled first-order differential equations. These are
dψ 4θ
=
(19)
dX F1
dθ
F
= 1 ψ 3/ 4
(20)
dX 3F2
subject to
X = 0, ψ = 0;
X = 1, θ = 1.
(21a.b)
Solving the foregoing equations numerically, Nader obtained the values of tip tempera-
ture, θ(0) for a range of values of F1 and F2. These values are recorded in Table 1.
The rate of heat conduction into the base of the fin, q can be obtained as
Table 1. Tip temperatures for a vertical rectangular
fin.
104
103
102
F1/F2
10
107
0.9970
0.9703
0.7460
0.0875
108
0.9947
0.9479
0.6011
0.0169
109
0.9905
0.9096
0.4182
0.0007
1010
0.9832
0.8461
0.2307
--
1011
0.9703
0.7460
0.0875
--
1012
0.9479
0.6011
0.0169
--
1013
0.9096
0.4183
0.0070
--
1014
0.8461
0.2307
--
--
1015
0.7460
0.0875
--
--
1016
0.6011
0.0169
--
--
1017
0.4183
--
--
--
8
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