Civil Engineering Reference
In-Depth Information
∂
q
x
∂
Integrating by parts on
x
with
u
=
N
i
and
d
v
=
d
x
,
we obtain, formally,
x
t
y
2
y
2
x
2
q
x
N
i
∂
q
x
∂
q
x
∂
N
i
∂
x
2
x
1
N
i
d
A
=
t
d
y
−
t
d
x
d
y
x
x
y
1
y
1
x
1
A
y
2
t
q
x
N
i
k
x
∂
T
∂
N
i
∂
x
2
x
1
d
y
=
t
+
d
A
(7.28)
∂
x
x
y
1
A
Now let us examine the physical significance of the term
y
2
y
2
q
x
N
i
x
2
x
1
t
d
y
=
t
[
q
x
(
x
2
,
y
)
N
i
(
x
2
,
y
)
−
q
x
(
x
1
,
y
)
N
i
(
x
1
,
y
)
]
d
y
(7.29)
y
1
y
1
The integrand is the weighted value (
N
i
is the scalar weighting function) of the
heat flux in the
x
direction across edges
a
-
a
and
b
-
b
in Figure 7.7b. Hence, when
we integrate on
y
, we obtain the
difference
in the weighted heat flow rate in the
x
direction across
b
-
b
and
a
-
a
, respectively. Noting the obvious fact that the
heat flow rate in the
x
direction across horizontal boundaries
a
-
b
and
a
-
b
is zero,
the integral over the area of the element is equivalent to an integral around the
periphery of the element, as given by
t
t
q
x
N
i
d
A
=
q
x
N
i
n
x
d
S
(7.30)
A
S
In Equation 7.30,
S
is the periphery of the element and
n
x
is the
x
component
of the
outward
unit vector normal (perpendicular) to the periphery. In our exam-
ple, using a rectangular element, we have
n
x
1
along
b
-
b
,
n
x
0
along
b
-
a
,
=
=
1
along
a
-
a
, and
n
x
0
along
a
-
b
. Note that the use of the normal vec-
tor component ensures that the directional nature of the heat flow is accounted
for properly. For theoretical reasons beyond the scope of this text, the integration
around the periphery
S
is to be taken in the counterclockwise direction; that is,
positively, per the right-hand rule.
An identical argument and development will show that, for the
y
-direction
terms in equation Equation 7.26,
t
=−
=
n
x
k
y
∂
N
i
d
A
t
∂
∂
T
k
y
∂
T
∂
N
i
∂
=−
q
y
N
i
n
y
d
S
−
d
A
(7.31)
y
∂
y
∂
y
y
A
S
A
These arguments, based on the specific case of a rectangular element, are
intended to show an application of a general relation known as the
Green-Gauss
theorem
(also known as
Green's theorem in the plane
) stated as follows:
Let
F
(
x
,
y
)
and
G
(
x
,
y
)
be continuous functions defined in a region of the x-y plane
