Deriving the thermal expansion formula for pipes

[dirk_mec1]

I do not understand how this formula is derived:

Can somebody help me?

Source:

https://www.corzan.com/en-us/piping-systems/specification/thermal-expansion

 

[JBA]

Note: Important information needed is missing from the above thread post.
It is necessary to go to the reference source and view its "loop" diagram consisting of two vertical legs' with a length ratio = 2L/5 of the calculated L requirement with two standard 90° pipe ells at the top connecting a horizontal pipe section with a leg to leg spacing ratio = L/5 requirement to understand what apparently results in the above formula.
As a result the above appears to be a formula based upon all of the above loop configuration requirements.

 

[dirk_mec1]

Note: Important information needed is missing from the above thread post.
It is necessary to go to the reference source and view its "loop" diagram

No that is not true! See here:

https://www.engineersedge.com/calculators/pipe_expansion_thermal_loop_15035.htm