Minimum Pipe Bore at the Erosional Limit

This is the same rule turned into a size. You know the flow and the mixture density; the erosional velocity follows from $C$ and the density, and continuity gives the smallest bore that keeps the velocity at or under it. Round up to the next commercial schedule and the line meets the specification.

The arithmetic is worth seeing in one line rather than two, because it shows how weakly the answer responds to anything. Combining $Q = V_e \pi d^2/4$ with $V_e = C/\sqrt{\rho_m}$ gives $d = \sqrt{4Q\sqrt{\rho_m}/(\pi C)}$. Diameter goes as the square root of flow, so doubling the throughput asks for only 41 % more bore. It goes as the fourth root of density — a factor of ten in mixture density moves the required diameter by 78 %. And it goes inversely as the square root of $C$, so the difference between the continuous 100 and the intermittent 125 is 11 % of diameter, which is often less than the step between two pipe sizes. That last observation is quietly useful in an argument: the choice between the two API constants frequently makes no difference at all to the pipe you end up buying.

Read the answer as a minimum against one criterion only. It says nothing about pressure drop, which for a long line usually governs and demands a larger bore than this. It says nothing about slugging or liquid holdup in two-phase flow, which is a flow-regime question and depends on inclination as much as velocity. And it says nothing about the minimum velocity needed to sweep solids and water along the bottom of the pipe. A line sized only against erosion can be too slow, and a slow line with sand in it develops a settled bed, a differential aeration cell under that bed, and a pit that goes through the wall while the average thinning rate looks fine.

In gas service there is one further wrinkle worth knowing. RP 14E's own guidance for gas lines expresses the limit as a minimum cross-sectional area rather than a diameter, because the mixture density in a gas line falls as pressure falls along the run, so the erosional velocity rises and the governing point is usually at the low-pressure end. Sizing on inlet conditions and forgetting the outlet is a standard way to build a line that passes the check on paper and fails it in the last few hundred metres.