1
NONLINEAR BOUNDARY VALUE PROBLEM SOLUTIONS
The temperature distribution of the rectangular fin shown in Figure 1(a) below, considering
conduction and radiation heat transfer, is given by
)
(
4
4
2
2
∞
−
=
T
T
kA
P
dx
T
d
σε
(1)
where
T
= temperature,
x
= location along the fin,
k
= thermal conductivity,
bd
A
=
= cross
sectional area,
)
(
2
d
b
P
+
=
= perimeter,
∞
T
= surrounding temperature,
σ
= StefanBoltzman
constant, and
ε
= emissivity. The values for the constants are given in the table below, where are
values are reported in a consistent set of units:
Constant
Value
Units
k
42
W/m – º
b
0.5
m
d
0.2
m
∞
T
500
º K
l
2
m
0.1
dimensionless
8
10
7
.
5
−
×
W/m
2
– º K
4
The boundary conditions are given as
1000
)
0
(
=
=
x
T
º K
350
)
(
=
=
l
x
T
º K
The temperature at three equally spaced interior node points 1, 2, and 3, as shown in Figure 1(b),
is to be determined using three different methods. This distance
h
between each pair of nodes is
0.5 m.
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a) Using Eq. (1), derive a nonlinear set of finite difference equations for the temperature at
interior nodes 1, 2, and 3. Use central differencing to approximate
2
2
dx
T
d
at each node.
Then write a Matlab program to solve the nonlinear system of equations for the temperature
at the three nodes. In your program, call the “fsolve” rootfinding function available in Matlab
using the default convergence tolerances and an intial guess of 675º K for all three nodes.
(Note: “fsolve” is only available in the Optimization Toolbox, so after coding your Matlab .m
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 Spring '09
 RAPHAELHAFTKA
 matlab, Numerical Analysis, Heat Transfer, Boundary value problem, finite difference, Finite difference method

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