Civil Engineering Reference
In-Depth Information
Owing to the algebraic volume of calculation required, examples of general
three-dimensional heat transfer are not presented here. A few three-dimensional
problems are included in the end-of-chapter exercises and are intended to be
solved by digital computer techniques.
7.7 AXISYMMETRIC HEAT TRANSFER
Chapter 6 illustrated the approach for utilizing two-dimensional elements and
associated interpolation functions for axisymmetric problems. Here, we illustrate
the formulation of finite elements to solve problems in axisymmetric heat trans-
fer. Illustrated in Figure 7.18 is a body of revolution subjected to heat input at its
base, and the heat input is assumed to be symmetric about the axis of revolution.
Think of the situation as a cylindrical vessel heated by a source, such as a gas
flame. This situation could, for example, represent a small crucible for melting
metal prior to casting.
As an axisymmetric problem is three-dimensional, the basic governing
equation is Equation 7.70, restated here under the assumption of homogeneity, so
that
k
x
=
k
y
=
k
z
=
k
, as
k
∂
2
T
T
2
∂
2
T
+
∂
+
∂
+
Q
=
0
(7.84)
x
2
y
2
z
2
∂
∂
Equation 7.84 is applicable to steady-state conduction only and is expressed
in rectangular coordinates. For axisymmetric problems, use of a cylindrical
coordinate system (
r
,
,
z
) is much more amenable to formulating the problem.
To convert to cylindrical coordinates, the partial derivatives with respect to
x
and
y
in Equation 7.84 must be converted mathematically into the corresponding par-
tial derivatives with respect to radial coordinate
r
and tangential (circumferential)
z
r
Figure 7.18
An axisymmetric
heat transfer problem. All
properties and inputs are
symmetric about the
z
axis.
q
z
