Analyse thermique et mecanique sous gradient radial.
Tube concentrique 2 materiaux, fluide chaud Ti=120C interne, paroi ext Te=20C.
Geometrie et proprietes
| Ri (mm) |
10 |
12 |
| Re (mm) |
12 |
20 |
| E (GPa) |
2000 |
2 |
| nu |
0,3 |
0,4 |
| k (W/mK) |
40 |
0,16 |
| alpha (1/K) |
1e-5 |
7e-5 |
Analyse thermique
Eq. chaleur stationnaire en coord cylindriques:
\[\frac{1}{r}\frac{d}{dr}(k r \frac{dT}{dr}) = 0\]
Solution: \[T_1(r) = T_i + \frac{C_1}{k_1}\ln\frac{r}{R_i}\] \[T_2(r) = T_i + \frac{C_1}{k_1}\ln\frac{R_m}{R_i} + \frac{C_1}{k_2}\ln\frac{r}{R_m}\]
\[C_1 = \frac{T_e - T_i}{\frac{1}{k_1}\ln\frac{R_m}{R_i} + \frac{1}{k_2}\ln\frac{R_e}{R_m}}\]
Expansion libre
\[u_r(r) = \alpha(r)(T(r)-T_0)r\]
Cavite rigide
\[u_r(R_e)=0\] genere pression de contact et contraintes.
import sys
sys.path.insert(0, "..")
import numpy as np
import matplotlib.pyplot as plt
from tube_params import TubeParams
p = TubeParams()
r = np.linspace(p.R_i, p.R_e, 500)
bg = (40/255, 43/255, 48/255, 1.0)
fig, ax = plt.subplots(figsize=(8,5))
ax.set_facecolor(bg)
fig.patch.set_facecolor(bg)
T_r = TubeParams.T_analytique(r)
ax.plot(r*1e3, T_r, "w-", lw=2)
ax.axvline(p.R_m*1e3, color="cyan", ls="--", alpha=0.5)
ax.set_xlabel("r (mm)", color="grey")
ax.set_ylabel("T (C)", color="grey")
ax.tick_params(colors="grey")
ax.set_title("Champ de temperature", color="grey")
plt.tight_layout()
plt.savefig("temperature_deplacement.png", dpi=150, bbox_inches="tight", facecolor=bg)
plt.show()
