Civil Engineering Reference
In-Depth Information
Table 7.1
Nodal Temperature Solutions
Four Elements,
Eight Elements,
T
(
◦
F)
T
(
◦
F)
x
(inches)
0
180
180
0.5
158.08*
155.31
1.0
136.16
136.48
1.5
123.59*
122.19
2.0
111.02
111.41
2.5
104.13*
103.41
3.0
97.23
97.62
3.5
94.01*
93.63
4.0
90.79
91.16
To illustrate convergence as well as the effect on gradient values, an eight-element solu-
tion was obtained for this problem. Table 7.1 shows the nodal temperature solutions for
both four- and eight-element models. Note that, in the table, values indicated by
*
are
interpolated, nonnodal values.
7.4 HEAT TRANSFER IN TWO DIMENSIONS
A case in which heat transfer can be considered to be adequately described by a
two-dimensional formulation is shown in Figure 7.5. The rectangular fin has
dimensions
a
t
, and thickness
t
is assumed small in comparison to
a
and
b
. One edge of the fin is subjected to a known temperature while the other three
edges and the faces of the fin are in contact with a fluid. Heat transfer then occurs
from the core via conduction through the fin to its edges and faces, where con-
vection takes place. The situation depicted could represent a cooling fin remov-
ing heat from some process or a heating fin moving heat from an energy source
to a building space.
To develop the governing equations, we refer to a differential element of a
solid body that has a small dimension in the
z
direction, as in Figure 7.6, and
examine the principle of conservation of energy for the differential element. As
we now deal with two dimensions, all derivatives are partial derivatives. Again,
on the edges
x
×
b
×
d
y
, the heat flux terms have been expanded in first-
order Taylor series. We assume that the differential element depicted is in the
interior of the body, so that convection occurs only at the surfaces of the element
and not along the edges. Applying Equation 5.53 under the assumption of steady-
state conditions (i.e.,
+
d
x
and
y
+
U
=
0
), we obtain
q
x
+
∂
d
x
t
d
y
q
y
+
∂
d
y
t
d
x
q
x
∂
q
y
∂
q
x
t
d
y
+
q
y
t
d
x
+
Qt
d
y
d
x
=
+
x
y
+
2
h
(
T
−
T
a
)d
y
d
x
(7.20)
