Highland Pipeline

What is modelled

A pump must lift water through a height Hs and also push it against friction in the pipe wall and in bends and valves. Friction head grows with the square of the speed, with the length, and inversely with the diameter, and the friction factor depends on the Reynolds number and the roughness of the wall. The pump head sets the electric power.

Equations

• Darcy-Weisbach hf = f (L / D) v² / (2 g), speed v = 4 Q / (π D²), Re = v D / ν.
• Colebrook-White 1 / √f = −2 log10(ε / (3.7 D) + 2.51 / (Re √f)); laminar f = 64 / Re.
• Pump head H = Hs + hf + K v² / (2 g) with K = 4, power P = ρ g Q H / η.

Checks run in the page

• Colebrook against Swamee-Jain, or Poiseuille flow integrated numerically when laminar.
• Wall shear stress balance against the friction head.
• Blasius smooth-pipe and von Karman rough-pipe limits.
• Flow split between two parallel pipes by bisection.

Honest simplifications

One straight pipe at a constant slope, water at 20 degrees Celsius, a lumped minor-loss coefficient, a constant pump efficiency and a steady flow. No surge or water hammer, no pump curve or system start-up, no tank level control and no energy tariff. The 3D scene is a greenhouse waterworks; this page is the pipeline arithmetic behind it, not a design tool.

• Press Narrow pipe, same flow and watch the friction head multiply.
• Press Old corroded pipe: a rough wall raises the friction factor although the flow is the same.
• Press Trickle (laminar-ish) and see roughness stop mattering.