Question 5.16: A pressure vessel operating at 300 °F is subjected to a shor...
A pressure vessel operating at 300 °F is subjected to a short excursion temperature of 600 °F. At a given time, the temperature distribution in the wall is shown in Figure 5.27. Find the maximum thermal stress at that instance. Let \mu=0.3, E=30 \times 10^{6} psi, and \alpha=6.0 \times 10^{-6} in./in.- °F.
This problem can be visualized as a biaxial case where the inside surface heats quickly while the rest of the wall remains at 300 °F. Using Eq. (5.29b) results in
\sigma_{x}=\sigma_{y}=-\frac{\alpha \Delta T E}{1-\mu} for a biaxial case. (5.29b)
\sigma=\frac{\left(6 \times 10^{-6}\right)(600-300)\left(30 \times 10^{6}\right)}{1-0.3}= − 77,100 psi,
which is extremely high and is based on very limiting assumptions. A more realistic approach is that based on Eq. (5.46). The mean temperature is obtained from Figure 5.27 and tabulated as follows:
\sigma_{\theta}=\frac{E \alpha}{1-\mu}\left(T_{ m }-T\right) (5.46)
|
Locations as ratios of thickness |
Temperature (°F) |
Area |
|
0 |
600 | |
| 0.1 | 460 |
53.0 |
|
0.2 |
400 | 43.0 |
| 0.3 | 370 |
38.5 |
|
0.4 |
340 | 35.5 |
| 0.5 | 320 |
33.0 |
|
0.6 |
310 | 31.5 |
| 0.7 | 305 |
30.8 |
|
0.8 |
300 | 30.3 |
| 0.9 | 300 |
30.0 |
|
1.0 |
300 | 30.0 |
|
\sum 355.6 |
And T_{ m } \approx 356^{\circ} F
From Eq. (5.46), at the inside surface,
\sigma_{\theta}=\frac{E \alpha}{1-\mu}\left(T_{ m }-T\right) (5.46)
\sigma=\frac{\left(30 \times 10^{6}\right)\left(6.0 \times 10^{-6}\right)}{1-0.3}(356-600)= −62,700 psi,
and at the outside surface
\sigma=\frac{\left(30 \times 10^{6}\right)\left(6.0 \times 10^{-6}\right)}{1-0.3}(356-300)= 14,400 psi.
It is of interest to note that the high stress occurs at the surface only.Thus at one-tenth of the thickness inside the surface, the stress is
\sigma=\frac{\left(30 \times 10^{6}\right)\left(6 \times 10^{-6}\right)}{1-0.3}(356-460)= −26,700 psi.
The high stress at the inside surface indicates that local yielding will occur.