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#### 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 TB at middle line (point B), external surface temperature TC (point C) and radial displacement wBR at middle line (point B) at times t1 = 10 s, t2 = 30 s, t3 = 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 = 2·1011 Pa;

Poisson’s ratio: µ = 0.3;

Density: ρ = 7850 kg/m3;

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/(m2·°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, % TB,1, °С 68.8037 68.54 0.38 TB,2, °С 306.193 306.19 0.001 TB,3, °С 380.001 380.09 0.023 TC,1, °С 46.2233 45.987 0.51 TC,2, °С 272.189 272.35 0.059 TC,3, °С 370.891 371.05 0.043 wBR,1, mm 0.0297 0.03155 6.229 wBR,2, mm 0.143 0.1452 1.54 wBR,3, mm 0.174 0.1744 0.23

Temperature distribution in pipe cross section at time t1, °С

Temperature distribution in pipe cross section at time t2, °С

Temperature distribution in pipe cross section at time t3, °С

Displacements distribution in pipe cross section at time t1, mm

Displacements distribution in pipe cross section at time t2, mm

Displacements distribution in pipe cross section at time t3, mm