Effects of Thermal Expansion
Temperature changes in the pipe has several effects. A homogeneous change in temperature will cause the pipe to extend or shrink in the axial direction. When there is a temperature gradient through the pipe wall, the mean temperature will control this deformation in an average sense. There will however also be local strain and stress states caused by the temperature variation. These stresses can be significant.
For analysis of cases with thermal expansion, it is assumed that the inside of the pipe has a temperature Ti, which is constant along the perimeter. As a first approximation, this would be the temperature of the fluid. In reality, there is a temperature jump given by the heat transfer coefficient. Similarly, it is assumed that the outside of the pipe has a temperature To, which is constant along the perimeter. Note that if there is insulation around the pipe, To represents the temperature of the pipe wall inside the insulation.
The variation of the temperature through the wall is treated slightly differently for circular cross section and in the general case. In the circular pipe, the temperature has a distribution which can be determined from analytically.
General Cross Section
For a general cross section, it is assumed that the temperature varies linearly through the pipe wall, and that the average temperature is
If the coefficient of thermal expansion is temperature dependent, it is assumed that its variation through the pipe wall is linear.
Define
The average axial thermal strain is then
Circular Pipe
For a circular pipe, the heat transfer problem can be solved analytically. The radial temperature distribution is given by
As long as the wall thickness is small, the temperature distribution through the thickness is close to linear.
If the coefficient of thermal expansion is temperature dependent, the exact axial strain has to be computed using an integral through the thickness of the material. In order to avoid performing such integration in runtime, a linear variation of the coefficient of thermal expansion with the radial coordinate is assumed,
The averaged axial thermal strain is then
Evaluation of the integral gives
Bending Thermal Stress
The temperature gradient through the pipe wall causes a local bending stress state, both in the axial direction and in the hoop direction. This contribution to the stress is taken into account only for the circular pipe section.
The peak value of the bending stress is calculated as