Civil Engineering Reference
In-Depth Information
PROBLEMS
7.1
For Example 7.1, determine the exact solution by integrating Equation 5.59 and
applying the boundary conditions to evaluate the constants of integration.
7.2
Verify the convection-related terms in Equation 7.15 by direct integration.
7.3
For the data given in Example 7.4, use Gaussian quadrature with four integration
points (two on
r
, two on
s
) to evaluate the terms of the stiffness matrix. Do your
results agree with the values given in the example?
7.4
Using the computed nodal temperatures and heat flux values calculated in
Example 7.5, perform a check calculation on the heat flow balance. That is,
determine whether the heat input is in balance with the heat loss due to convection.
How does this check indicate the accuracy of the finite element solution?
7.5
Consider the circular heat transfer pin shown in Figure P7.5. The base of the pin
is held at constant temperature of 100
◦
C (i.e., boiling water). The tip of the pin
and its lateral surfaces undergo convection to a fluid at ambient temperature
T
a
.
The convection coefficients for tip and lateral surfaces are equal. Given
k
x
=
380
W/m-
◦
C,
L
2
cm,
T
a
=
30
◦
C
.
Use a two-
element finite element model with linear interpolation functions (i.e., a two-node
element) to determine the nodal temperatures and the heat removal rate from the
pin. Assume no internal heat generation.
2500
W/m
2
-
◦
C,
d
=
8
cm,
h
=
=
h
,
T
a
h
,
T
a
100
C
L
Figure P7.5
7.6
Repeat Problem 7.5 using four elements. Is convergence indicated?
7.7
The pin of Figure P7.5 represents a heating unit in a water heater. The base of the
pin is held at fixed temperature 30
◦
C. The pin is surrounded by flowing water at
55
◦
C. Internal heat generation is to be taken as the constant value
Q
25
W/cm
3
.
All other data are as given in Problem 7.5. Use a two-element model to determine
the nodal temperatures and the net heat flow rate from the pin.
=
7.8
Solve Problem 7.5 under the assumption that the pin has a square cross section
1cm
×
1cm
.
How do the results compare in terms of heat removal rate?
7.9
The efficiency of the pin shown in Figure P7.5 can be defined in several ways.
One way is to assume that the maximum heat transfer occurs when the entire pin
is at the same temperature as the base (in Problem 7.5, 100
◦
C), so that
convection is maximized. We then write
L
q
max
=
hP
(
T
b
−
T
a
)d
x
+
hA
(
T
b
−
T
a
)
0
where
T
b
represents the base temperature,
P
is the peripheral dimension, and
A
is cross-sectional area at the tip. The actual heat transfer is less than
q
max
, so we
