Civil Engineering Reference
In-Depth Information
Using the result of Example 7.8, the stiffness matrix for a one-dimensional
heat transfer element with conduction, convection, and mass transport is given
by
1
21
12
−
k
(
e
)
=
k
x
A
L
hPL
6
mc
2
˙
−
1
11
+
+
−
11
−
11
=
k
(
e
c
+
k
(
e
h
+
k
(
e
)
m
(7.69)
where the conduction and convection terms are identical to those given in Equa-
tion 7.15. Note that the forcing functions and boundary conditions for the one-
dimensional problem with mass transport are the same as given in Section 7.3,
Equations 7.16 through 7.19.
EXAMPLE 7.9
Figure 7.16a shows a thin-walled tube that is part of an oil cooler. Engine oil enters the
tube at the left end at temperature
50
◦
C
with a flow rate of 0.2 kg/min. The tube is sur-
rounded by air flowing at a constant temperature of
15
◦
C
. The thermal properties of the
oil are as follows:
=
0
.
156 W/(m-
◦
C)
Thermal conductivity:
k
x
Specific heat:
c
=
0
.
523 W-hr/(kg-
◦
C)
The convection coefficient between the thin wall and the flowing air is
h
=
300 W/(m
2
-
◦
C)
. The tube wall thickness is such that conduction effects in the wall are to
be neglected; that is, the wall temperature is constant through its thickness and the same
as the temperature of the oil in contact with the wall at any position along the length of
the tube. Using four two-node finite elements, obtain an approximate solution for the tem-
perature distribution along the length of the tube and determine the heat removal rate via
convection.
■
Solution
The finite element model is shown schematically in Figure 7.16b, using equal length
elements
L
=
25 cm
=
0
.
025 m
. The cross-sectional area is
A
=
(
/
4)(20
/
1000 )
2
=
Air, 15
C
T
50
C
20 mm
m
m
1
2
3
4
5
100 cm
1
2
3
4
(a)
(b)
Figure 7.16
(a) Oil cooler tube of Example 7.9. (b) Element and node numbers for a
four-element model.
