Head Loss in Pipes

Tap a level, constant-diameter pipe at intervals and the total head falls steadily along it. No work was extracted; the energy went to heat through viscous shear near the wall. The temperature rise is unmeasurably small, and the pressure it cost is exactly what a pump has to replace.

Dimensional analysis forces the loss into the form h_L = f·(L/D)·(v²/2g) — a dimensionless friction factor, times slenderness, times velocity head. All the physics that resists analysis is packed into f.

Diameter is the brutal one. It appears twice — once in L/D and once through continuity in v — which is why pipe sizing dominates every hydraulic design and why a single narrow section can cost more head than the rest of the run combined.

For laminar flow the velocity profile is exactly parabolic, and integrating the wall shear gives f = 64/Re — a genuine derivation, no chart required. Substituting back makes head loss proportional to v rather than v², which is the Hagen-Poiseuille result.

Roughness does not appear anywhere in the laminar result. The slow layer at the wall never reaches the bumps, so a rough pipe and a smooth one lose the same head.

For turbulent flow there is no closed solution, so f is measured. The Moody chart plots it against Reynolds number for a family of relative roughness values ε/D. The Colebrook equation is the algebraic fit — implicit in f, so it is solved by iteration or with an explicit approximation such as Swamee-Jain.

At high Reynolds number the viscous sublayer thins until roughness elements protrude through it, and the curves on the Moody chart go flat. f then depends on ε/D alone and stops moving with flow rate.

Most large water mains and oil pipelines run here, and it is a practical gift: the friction factor holds steady across the demand range, so one number sizes the pipe rather than a fresh calculation for every operating condition.

Every bend, valve, entry, exit and change of section costs h_L = K·v²/2g, with K from tables. The name is a historical accident.

In a long transmission main these genuinely are minor. In a plant room dense with fittings they routinely exceed the straight-pipe loss — and a throttled valve can exceed everything else combined, which is why throttling to control flow is thermodynamically wasteful compared with varying pump speed.

Add static lift to the computed losses and you have the system curve: head required against flow rate, rising roughly as Q². Overlay the pump's own curve, which falls with flow, and they cross at exactly one point.

That intersection is where the installation will run, whatever anybody intended. It is the only stable point: below it the pump delivers more head than the system needs and the flow rises; above it the reverse.

Pipe diameter trades capital cost against a pumping bill paid for decades, so the optimum is an economic calculation rather than a hydraulic one. Undersize it and the energy cost runs for thirty years.

The suction side has a hard limit. If losses drop the pressure to the liquid's vapour pressure, vapour bubbles form and then collapse violently where the pressure recovers — on the impeller. That is cavitation: it sounds like gravel, it erodes metal, and it is prevented by keeping NPSH available above NPSH required.