Civil Engineering Reference
In-Depth Information
q
y
q
y
d
y
y
q
h
d
y
Q
,
U
q
x
x
q
x
q
x
d
x
b
T
q
h
q
h
d
x
t
h
(
T
T
a
)
q
y
a
Figure 7.5
Two-dimensional
conduction fin with face and
edge convection.
Figure 7.6
Differential element depicting
two-dimensional conduction with surface
convection.
where
t
=
thickness
=
the convection coefficient from the surfaces of the differential element
T
a
=
the ambient temperature of the surrounding fluid
Utilizing Fourier's law in the coordinate directions
h
k
x
∂
T
q
x
=−
∂
x
(7.21)
k
y
∂
T
q
y
=−
∂
y
then substituting and simplifying yields
t
d
y
d
x
t
d
y
d
x
∂
∂
k
x
∂
T
∂
∂
k
y
∂
T
T
a
)d
y
d
x
(7.22)
where
k
x
and
k
y
are the thermal conductivities in the
x
and
y
directions, respec-
tively. Equation 7.22 simplifies to
Qt
d
y
d
x
=
−
+
−
+
2
h
(
T
−
x
∂
x
y
∂
y
tk
x
∂
tk
y
∂
∂
∂
T
∂
∂
T
+
+
Qt
=
2
h
(
T
−
T
a
)
(7.23)
x
∂
x
y
∂
y
Equation 7.23 is the governing equation for two-dimensional conduction with
convection from the surfaces of the body. Convection from the edges is also
possible, as is subsequently discussed in terms of the boundary conditions.
7.4.1 Finite Element Formulation
In developing a finite element approach to two-dimensional conduction with
convection, we take a general approach initially; that is, a specific element geom-
etry is not used. Instead, we assume a two-dimensional element having
M
nodes
