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.