Pipe flow & pressure drop

Darcy–Weisbach

Δp = f · (L/D) · ½ρU²

Minor losses use the same dynamic pressure with a coefficient instead of a length ratio, Δp = ΣK · ½ρU², and the static term is just ρgΔz. Nothing here is subtle. The friction factor is where the work is.

The friction factor

Below Re = 2300 the flow is laminar and the answer is exact:

f = 64 / Re

Above it, Colebrook–White is implicit in f:

1/√f = −2 log₁₀ ( ε/3.7D + 2.51/(Re√f) )

Most calculators substitute the Swamee–Jain explicit fit and stop. This one uses that fit as a seed and then iterates the real equation to convergence. The difference is under two per cent for ordinary pipe, but it costs nothing and it means the number does not drift in the rough-turbulent corner where the explicit fits are weakest.

Velocity is the real design variable

Pressure drop goes with the square of velocity, so a line one size up costs far less pumping energy for its whole life. The usual liquid band is roughly 1 to 3 m/s: below about 0.6 m/s you stop sweeping air and solids along, above about 3.5 m/s you start buying erosion, noise and water hammer. The tool flags both ends.

Between 2300 and 4000

There is no reliable friction factor in the transitional band. If your duty point lands there, do not tune the calculation — change the pipe size so it does not.