Civil Engineering Reference
In-Depth Information
Substituting numerical values into Equation 7.49,
12(20)
4(0
.
5)
(89
q
(2)
x
84 Btu/(hr-ft
2
)
=−
.
041
+
90
.
966
−
106
.
507
−
111
.
982)
=
4617
.
12(20)
4(0
.
5)
(90
q
(2)
y
00 Btu/(hr-ft
2
)
=−
.
966
+
111
.
982
−
106
.
507
−
89
.
041)
=−
888
.
and, owing to the symmetry conditions, we have
q
(3)
x
84 Btu/(hr-ft
2
)
=
4617
.
q
(3)
y
00 Btu/(hr-ft
2
)
=
888
.
as may be verified by direct calculation. Recall that these values are calculated at the
location of the element centroid.
The element resultants representing convection effects can also be readily
computed once the nodal temperature solution is known. The convection resul-
tants are of particular interest, since these represent the primary source of heat
removal (or absorption) from a solid body. The convective heat flux, per Equa-
tion 7.2, is
T
a
) Btu/(hr-ft
2
)orW/m
2
(7.52)
where all terms are as previously defined. Hence, the total convective heat flow
rate from a surface area
A
is
=
−
q
x
h
(
T
H
h
(7.53)
=
h
(
T
−
T
a
)d
A
A
For an individual element, the heat flow rate is
H
(
e
)
h
h
(
T
(
e
)
(7.54)
=
−
T
a
)d
A
=
h
([
N
]
{
T
}−
T
a
)d
A
A
A
The area of integration in Equation 7.54 includes all portions of the element sur-
face subjected to convection conditions. In the case of a two-dimensional element,
the area may include lateral surfaces (that is, convection perpendicular to the plane
of the element) as well as the area of element edges located on a free boundary.
EXAMPLE 7.7
Determine the total heat flow rate of convection for element 3 of Example 7.5.
■
Solution
First we note that, for element 3, the element-to-global correspondence relation for nodal
temperatures is
T
(3)
1
T
(3
4
=
[
T
5
T
(3)
2
T
(3)
3
T
8
T
9
T
6
]
