Katelyn,
Does the procedure have a signature sheet on the front from which you can track down the original author? Is there a list of references or bibliography that you can track from footnote numbers? You might then be able to ask where the equations come from. I have not attempted the derivations myself, but on first glance the equations appear to be (and perhaps indeed ought to be) the result of combining the effects of circumferential and longitudinal pipe strain due to pressure and temperature rise with the effects of the compressibility of water. Probably quasi-theoretical and quasi- empirical. I suspect that the "restrained" case should depend to some extent on soil conditions and virtual anchor length, which will vary from place to place.
I view it is unfortunate that equations of this nature find their way into documents of this type but that the source of these equations gets lost. It is my belief that the author(s) ought to provide either the sources or the derivations. When I have written certain documents or reports that include an equation for something in support of the text, I always include an appendix that shows either the source or the actual derivation. Otherwise, you end up in a situation like the one you are in: a bright engineer asking "Where did this come from?" in a company where nobody knows the answer. That produces, over time, a "Design By Xerox" mentality. People get psyched into thinking, "Ooooh...look at that equation...it's complicated and hard...it *must be* right...it *must have* come from really smart people and lots of science.". (much like the API 14E correlation for erosion velocity limits, I imagine). And, well, maybe it did, or maybe it didn't, or maybe it's wrong. What I will offer you is this: good on ya for checking it thoroughly in an effort to understand it.
How much time are they providing you with for this task and to what level of detail do they want you to check it out? If you have a couple of days, you could prepare your own spreadsheet that compares various methods: the one you have in the procedure versus the one proposed by zdas04 versus your own first-principles derivation versus those that you might obtain from textbooks that deal with piping and pipelines versus those that might be published in Codes and Standards versus those that are used by pipeline construction company guidelines. If they compare closely, you then have succeeded in verifying with reasonable level of comfort that the equations in the procedure are valid. Or, you might find yourself in a position where you can conclude that the much simpler correlation proposed by zdas04 is valid and you will have the pages of your validation effort to prove it.
I recently wrote a technical report in which I came up with a calculation methodology to quantify the prediction of erosion rates ceased by entrained frac (and / or formation) sand in flowing gas wells. Basically, the end result was something incredibly simple: if [number A] is this, go up one nominal pipe size; if [number A] is this, go up two nominal pipe sizes; otherwise install [facilities]. I got there after reading fourteen papers, reading two failure reports, reading a couple of textbook chapters, creating a custom spreadsheet and performing 40 pages of calculations - but I got there. A colleague for whom I have great respect and who holds a PhD in chemical engineering, said that the simpler the answer, the better the guideline. Keep the calculation records to be produced on request, but meanwhile, only publish the little table of [number A] versus [do this].
In other words, in your position, I would make it as simple as possible, and maybe tie it back to a reference document that, if necessary, captures the proof of why it boils down to something that simple. You will only need to do that once or twice before people start saying, "Holy crap is this person smart.".
