TEST CASE 12.2 TERMAL EXPANSION IN THICK-WALLED PIPE CROSS SECTION CAUSED BY NON-UNIFORM HEATING
Reference:
Numerical solution (Abaqus).
Problem description:
Heat carrier is delivered into the pipe in such a way that temperature T of inner side varies in time, as show at figure below. Initial pipe temperature before heat carrier inlet is 20 °С. Convective heat-exchange interacts at external side. Find temperatures T_{B} at middle line (point B), external surface temperature T_{C} (point C) and radial displacement w_{BR }at middle line (point B) at times t_{1} = 10 s, t_{2} = 30 s, t_{3} = 50 s respectively.
Problem sketch:
Type of created problem:
Spatial structure (X, Y, Z, UX, UY, UZ).
Geometry properties:
Outer radius: R = 50 mm;
Inner radius: r = 30 mm.
Material properties:
Elastic modulus: E = 2·10^{11} Pa;
Poisson’s ratio: µ = 0.3;
Density: ρ = 7850 kg/m^{3};
Expansion coefficient: α = 1.2·10^{-5} 1/°С;
Conductivity: λ = 60.5 W/(m·°С);
Specific heat: С = 434 J/(kg·°С);
Heat transfer coefficient: β = 60 W/(m^{2}·°С).
Boundary conditions:
Z = 0;
Symmetry boundary conditions.
Model description:
System is modeled via 576 4-noded shell elements (FE type is 44) and 36 bar elements of surface heat transfer (FE type is 168). Due to symmetry, only quarter of pipe section with corresponding boundary conditions is considered. Dynamic calculation with heat exchange is performed (full time is 50 s, step time is 0.2 s).
Calculation results:
Target value |
Abaqus solution |
LIRA 10 |
Deviation, % |
T_{B,1}, °С |
68.8037 |
68.54 |
0.38 |
T_{B,2}, °С |
306.193 |
306.19 |
0.001 |
T_{B,3}, °С |
380.001 |
380.09 |
0.023 |
T_{C,1}, °С |
46.2233 |
45.987 |
0.51 |
T_{C,2}, °С |
272.189 |
272.35 |
0.059 |
T_{C,3}, °С |
370.891 |
371.05 |
0.043 |
w_{BR,1}, mm |
0.0297 |
0.03155 |
6.229 |
w_{BR,2}, mm |
0.143 |
0.1452 |
1.54 |
w_{BR,3}, mm |
0.174 |
0.1744 |
0.23 |
Temperature distribution in pipe cross section at time t_{1}, °С
Temperature distribution in pipe cross section at time t_{2}, °С
Temperature distribution in pipe cross section at time t_{3}, °С
Displacements distribution in pipe cross section at time t_{1}, mm
Displacements distribution in pipe cross section at time t_{2}, mm
Displacements distribution in pipe cross section at time t_{3}, mm