Civil Engineering Reference
In-Depth Information
Applying Equation 7.89 to the operations indicated in Equation 7.90 yields (with
appropriate use of trigonometric identities)
2
T
2
T
2
T
∂
2
T
∂
∂
+
∂
∂
1
r
∂
T
∂
r
2
∂
1
(7.91)
=
+
r
+
∂
x
2
∂
y
2
r
2
2
where the derivation represents a general change of coordinates. To relate to an
axisymmetric problem, recall that there is no dependence on the tangential coor-
dinate
. Consequently, when Equations 7.84 and 7.91 are combined, the gov-
erning equation for axisymmetric heat transfer is
k
∂
2
T
∂
2
T
1
r
∂
T
∂
r
+
∂
+
+
Q
=
0
(7.92)
r
2
z
2
∂
and, of course, note the absence of the tangential coordinate.
7.7.1 Finite Element Formulation
Per the general procedure, the total volume of the axisymmetric domain is dis-
cretized into finite elements. In each element, the temperature distribution is
expressed in terms of the nodal temperatures and interpolation functions as
M
N
i
(
r
,
z
)
T
(
e
)
T
(
e
)
=
(7.93)
i
i
=
1
where, as usual,
M
is the number of element nodes. Note particularly that the
interpolation functions vary with radial coordinate
r
and axial coordinate
z
.
Application of Galerkin's method using Equations 7.92 and 7.93 yields the
residual equations
k
∂
Q
N
i
r
d
r
d
2
T
2
T
∂
1
r
∂
T
r
+
∂
+
+
d
z
=
0
i
=
1,
...
,
M
r
2
z
2
∂
∂
V
(7.94)
Observing that, for the axisymmetric case, the integrand is independent of the
tangential coordinate
, Equation 7.94 becomes
k
∂
Q
N
i
r
d
r
d
z
2
T
2
T
∂
1
r
∂
T
∂
r
+
∂
2
+
+
=
0
i
=
1,
...
,
M
r
2
z
2
∂
A
(
e
)
(7.95)
where
A
(
e
)
is the area of the element in the
rz
plane. The first two terms of the
integrand can be combined to obtain
k
1
r
r
∂
Q
N
i
r
d
r
d
z
2
T
∂
∂
∂
T
+
∂
2
+
=
0
i
=
1,
...
,
M
z
2
r
∂
r
(7.96)
A
(
e
)
